The American Term Life Insurance History Project

You are reading a page from An Elementary Treatise on Actuarial Mathematics by Harry Freeman (1932)
Part of the American Term Life Insurance History Project
Term Life Insurance

ANSWERS TO THE EXAMPLES
Examples 1.

i A/3 1 I I 1 V3 1 i A^/3 1
₂,— ₂,-A/3' 1^/2^' V2> _ _I; 2' 2 , ~₃;^-₂,^-₂, _1/35r 377 25r 7^r(v) nr+4.

8.(i) ⁷; (ii) ⁷; (iii) r; (^iv) 2^r; (^v) ?T; (vi) --; (vii) 4'r;

(viii) — ; (ix)

27r;

(x)

9.(I) IS^; ^(II)22' (ⁱⁱⁱ) g; (Iv) 52 ; (^v) (3ⁿ —I) L^T.

10.— cos 0; — sin 8; — tan 0; tan 0; cosec 0; — cosec 0; — cot 0; sin B; sin 0. 5⁶. ¹3.19²⁹₂₄₀³5⁶. ²

12.

65'

⁸

3 A/5

₅

' 289' 5' 33'

(4n + 1) or (2n + 3) ⁷₂ + tari^—1

34.

(2n — i) 7r — a.

35.

1 +

^'n

36,

7177 + (— I)

ⁿ

sin —i

₄

^{3 i}

•

₄

s .

Solving in the ordinary way, x = 2. On substitution, however, x = 2 gives a positive value to the left-hand side of the equation, whereas the right-hand side is negative. Strictly speaking, therefore, there is no solution.

2,17r ± 2 , 4n7r ± r or - (4n7r ± 7r).40. x = 4 •

₃

48. ± — .

2. -_{6' 4'}₆ ' ₆.

(a) 6 ; (b) 3 ; (^c) 6 ; (^d) 4; (e) o.

(a) 1-008; (b) 1^.006; (c) 2^.233^.

7. (i)

n7r + (— I)

ⁿ

₄

;

(ii) 2nr

⁷

;

(iii)

(iv)

2nrr +

₄

;

46. {

in + } r

or {

n — k}

49.

_{a=inr+(-1)n4r;}

$=:nn+(-

^rz

•

5o,

sin {

(n—1)

sin

in3

sin'

3⁸4    ACTUARIAL MATHEMATICS Examples 2.
1. 58.    2. 30, 42.    3. 15.
4. ^1.9, 4'9.    5. 11 lo.    8. 6ah³.
9. A (— 11x³ + 252X² — 1051X + 1344).    10. abcd. Io!.

ab" (b^e-1); ab^ex(b°-1)²; ab`x-(b°-I) be -z.

I)'

-I]

(i) Ix (x — ,) + k,(ii) c^x/(c — I) + k,

(iii) 3X

(x -

(x - 2)

(x -

3X +

where

is a constant.

2_3

-(x+2)(x+3) (x+3)(x+4)'

(x+ 2)

(x+3)(x+4)+(xt

3)(

+4)(

x(x-1)(x-2)

16.

(1)

an!;

(2)

eax+b

_(e

—

1)n.

18. 55

Ix (x — 1) (x — 2) (x — 3) + 2x (x — 1) (x — 2) + x (x — 1) + I2x + k.

2225.22. 20.23, — 161.24. 229.25. 1261.

x(⁴) - 6x⁽³⁾+ 13x⁽²⁾+ x⁽'⁾+ 9; 4x(3) - 18x⁽²⁾+ 26x⁽'⁾ + 1;

I2x

⁽²⁾

36x

⁽

⁾

+ 26; 24x

⁽

⁾

⁶

;

24.

(i) (m + 1)m(m-1)...(m-n +2) a^m+' (b/a+x+m)(m-"+');

(ii) (—

(m +

(m + z) ... (m + n) a

""" (b/a + x)

^(-m+

"+')

z cos (x + la) sin la ; sin a/ {cos (x + a) cos x} ; a - 2 sin (x + a) sin la.

6x; 6/(x + 1)².42. y²+ 4ay = /3².43. 2 (x - 2)" - 2 (x - 3)".

15.

-2or109.

19. _a2x
+ (a² + I)2 a4x_.
15.

1. 465.
4. 182; 343^. 7. 128.
10. 97,357^. 13. 69,215. 17. 14^.73658. 20. 5479^.

22. 23. 24. 26. 29.

31. 32.
35.

Examples 3.

44¹; ⁶53^.54¹4^.
94; 39⁶; 66z. ⁸44; 74⁶•
3. 300.
6. ⁸9,9²⁰; ⁸9,⁰73^. 9. ¹94'3; ²79'9^. 12. '98127.
'432; -^.33⁸; -^.19^6. 19. 5281; 6504.
²'37²²3^. 3^.708; 3^.711.
21. ²¹53; ¹7⁰5^.²459; ²4²4; 2359; 2268; 2153; 2018; 1868; ¹7⁰5; ¹534; ¹357; '⁰¹7; '⁰35; '052; •070; '087; '104; '122; •¹39; '¹57^.²3'¹²34; 23'2039; 23^.2914; 23'3865; 23^.4898; 23^.6019; ²3'7²34^.
I•000. 27. •020660; '020625; '020628. 28. 58,844. 1; 2^.10; 3'31; 4'64; 6'II; 7'73; 9'5¹; ¹¹'47; ^{13.62; 1}5^.97^.117^.7; 114^.2; II0^.5; 106^.7; Io2.7; 9⁸'⁶; 94'3; 89'8; 85'2; ⁸⁰'3; 75'4^.
'24928. 33. Third degree: ²75^.^34.459.
1[₂⁼218; u_i⁼o; ay^=—19; ux=1876-1429x+36ox²-3ox³.

I180.

ANSWERS TO THE EXAMPLES Examples 4.

2. _47,983.3. 2'8169. 4. _1.7243. 5.

1.
5745^.

3⁸5 2300.    6. 460.
_{7 -}1+m lm+mn+nl _—lmn+mnp+npl+plm
1²m²'    1²m²n²    1²m²n²p²8. 13^.18.    9. 14^.942.    10. 20^.43.    11. 162.
12. 659+224x+82x²— ;¹.,x³.    13. 32.    16. I; 25.
18. 33 and 67 to the nearest integer.    19. 37^.2.
20. _7.37.21. 130,326.

Examples 5.

1. 33.    2. 6.    ³. 47,692.    4. 3251.    5. 16^.9216.
6. 2^.85805; 2^.86305; 2^.86157; 2^.86155.    7. 2017.
8. 3^.5283.    9. 2196,2108,2022,1939; 1786, 1718, 1657, 1604.
10. •01625.    11. 3165.    12. 2290^.1.    14. 4'⁰34^.

Examples 6.

1. 471^.5; 2^.7. 2. 13 3. 3. 2'019...; 2^.018....
4. 43^.1. 5. 8'34. 6. _2.751.7. 16^.9.

8.I^.1576....9. 1^.2134.10. 45^.70.11. 1 85 .

12.

091.13.

667

per cent.14.

3713.

15. 37

Examples 7.

4n (—

27n

17).

₁

₂

+ 1)

(3n

+ 7n

2).

3. —

4195.

²¹

+ 628.

3n+1

(

+ 7n — 6).

_A{

ⁿ

—

+ 5n

An (n

(2n

+ I)).

₂

21:+1 —

2 —

k (2k

+ I)

(4k

+ I).

(

⁴

—

Ion

Ion).

9.2n⁴ + i6n^a + 47n² + 6on.10. 2¹⁹— ²⁰95^.

11.

4(n+3)(n+4)(n+5)(n+6)—90.

12.

in(n+

(n +

(n + 5).

13.{

(3n —

(3n

+ I)

(3n +

(3n + 7) + 56}.

14.

₁

+ I)

(3n

13).

15.

₁

n (n +

i) (n

2) (6n

+ 57n

137).

16.

-,.A

_l{

(2n + 3) (2n +

(2n + 7) (2n + 9) (2n +

II) —

^1o

395}

ⁿ

_13)

n(n+

4(n+4)

18.

12(n+2)(n+3)'

19.

6(n+3)(n+4)•

17.

386    ACTUARIAL MATHEMATICS
n(3n+5)    n(5n+ II)
20. ⁸(3ⁿ+ 1)(3n+4)'    21. 4(n+ 1)(n+2)^,
22. H n (n + I) (n + 2) (3n² + 36n + Ioi). 23, 19 –    12n² + 33n + 19    19
168. 6 (3n + I)(3n + 4) (3n + 7)^' 168^'

(n + 1) (3n² + 31n + 74)^.

+(n+3)(n+2)(n+

in(n

–

I)(n–2)(n–3).

26.— r —

27. -

(

_x2

—

a — tax

(aa (a +

I)l

_C.

a11

-1)

I — 3 (n + I) ! + (n + 2) !.

12 {

(3n –

(3n

2) (3n + 5) (3n + 8) (3n

11)

+ 88o}.

n (6n³ + 16n² + 9n – 4).31. ₂? n (7n³ – 34ⁿ² + 89n – ²54)^.

_12n(n + I) (9n² + 17n + 4); {2^1}1(n³ – 3n² + Ion – 14)) + 28.

3 + 4n ₃+1 {n +_ z} .35. _768'36. 34 – (4ⁿ²+ Ian + 17)/2ⁿ--¹.

37. (x –

2) 3

38.

–

1) + 1

(42n +

17n

+ n

⁴

39.

42.

40.

(n+2)x

ⁿ

—

^"t+1

—x

x —

(x (x —

₃₉

36n + 39

¹⁰

ⁿ

+z)(3

⁷¹

+5)1•

I2x

+ 36x

28x _+ 3

12x

+ 1)

2) (x + 3) '

43.

23.

44.

₂

n+1

45. z

—

2 {

2 — (n + 2)

ⁿ}

2.x2

(I — x").

^—

C – 2x-2 (x – 11 (2x– I)!'

+_$¹-₆(2n– 1)(zn+ 1)(2n+3)(24n+54n+25).

ax' – (la –

3b)x

+(ea–

lb+ c)x

+(tb–

is+d)x+K.

(I —

^r+1

Examples

20'796875; – u i •

10.

_5'254.

^13.

⁰

14.

319.

15. 3

9634.

16.

^10.

⁸

¹⁰

'475

Examples

₄

^n-Q

3.(log

.5. 1 (a — k).7. o.

.9.

10.

11.

12. e

(log

—log

^13.

14.

15.

16. oo .

17.

20.

46.

3⁸7
ANSWERS TO THE EXAMPLES
Examples 10.

z
5. i/a2 — x² — x ; ex log x (log x + 1). v'_az — x²
6. 5X⁴; an (ax + b)'°^-1; x^x(I + log x).
n

7.bn (a + bx)"^-1; na1^x log a; zx/n (a² + x²) n ; x - log a.

—

_x2

;

^n—1

(I —

^n—I

(m — m +

nx) ; x'

^—1

(x +

m) ;

^1-1

+ n log x).

^9'

—

log

x (1

log

x);

^10x

(log

Io)

lox;

log

_lo

;

2 sin x cos x; 2 cos 2x; — 3 cos^t x sin x; x sec x (2 + x tan x).

2X; sin x + x cos x; sec^t x tan 2x + 2 sec^t 2x tan x.

—

2 VX

sin x

12.

cos'

;

tan

^—1

x + -

₁

_x2

;

VI—x

cos

(tan

^—1

x)2 +

sin

tan

+ x

(5 + 4^x) ^I°^g {x ^log(5 + 4X) + ₅+ 4x log xl

—

mx'°

^-1

I -

² {

I - V

I -

²}

log x

a^x log a + ax°^-1; x^x (I + log x) - 2mx (I - x²)"°-¹;

xxn

^ri-1

log

x + I).

³^(I—4^x2). 3 vx Vcos^-I x — ^x.-^x/

(1 + x

) (I +

16x

)

2 V

I -

cos

y2I

18. x - xylogx'

^19.

x e

- 1

21. (i)

⁴

;

_(ii)

)

⁼

)}

2x (a + bx

ⁿ

)}

- _-4^x jtan_1 2x^z + a _I-I

+ I (2ax

- J

(i) _ex xer {log x + x) ; (ii) {log 1 + x1 + x} ^{if I}-

26.

3 (log x)

flog

27.

—

)

—

(I +

)

23. (¹+ t²)²' (I + t²)^2'(I - t²)³' (I + t²)
20.    feet per second.
25-2

3⁸⁸29.
ACTUARIAL MATHEMATICS
2e^x
32.
ve2x —    _a2.
34. (i) cot x; (ii)     4 + x²'
a sin a/x — cos a/x 2x² ' I + sin a/x cos a/x

2x4 + _y2
_x3

39. (i) ^exxxx(I + log x); (ii) exsxx (I    log x); (iii) e^xz.
    (i) ₆₂(ad — bc)    _(ii)I I — 6 log x
    (cx + d)⁴'    x⁴
44. 4x2 — 4x.    46. 2 log a •    50. o. a
51.    n l    pa+q    i    pb+q
(— I) n. ~(a — b) (a — c) (x — a)ⁿ+ⁱ + (b — c) (b — a) • (x
1
pc + q
-    I
+(c—a)((c—b)(x—c)n+i}
    (— _I)n n! !_I-      ⁶     — --- ²
l(^x — z)n+1 (^x— I)ⁿ⁺¹(— i)ⁿn! {(c—a)(c — b)_(d—a)(d—b)
    c — d    (x — c)ⁿ+ⁱ    (x — d)n+1 }
    8    256
54. (I — x)⁵ + (I + 2x)⁵
r
57. (i) (— I)',-1 ?(n-- 3)!. (ii) (— I)ⁿ n!}^I    ^5.3f+    - ^Ix^n—2'    l(3^x + )n+1 (x — )n}11•
59. a^ri}²x²e"^x.
(n + I)!
(— I)ⁿ (b ⁿ a)2 _((x—^Ib)n+1^— (x — a)n+1} — (— I)ⁿ (b — a) (x — _a)n+2•
70. p = (log s)² ; q = log c log g (log cs²) ; r = (log g log c)².
—cos0    r
(i) N^'cos 20—cosec² B cos² 28 (ii) 2 (I +xx²
_3.74. (o, o) and (za, — 4a³/3b²).
(ⁱ) (— 2, 3) (— 2, — ^I); (ⁱⁱ) (^I, ^I) (— 5, ^I)^.

Examples 11.

x _x2    x4    5
    2. log2+-+---    ...
³• I + x z — 2+
    2    8    192
₆    x x² x³x²x³x'    x⁵    x⁶    6. I + —    +    ....    7. I + 2x + - + +    + - + -
    -    ₁ 2    24    ... .
2    6 12 40 720
I    3a²x² + 2ax³ + x⁴

30.

33. 35. 37.

3 (a + x) (a² + x²) (a² + ax + x²)

I I+ VI—x⁴
2x² loge I o' x6 — -- .

I — 2Xy
x²+ 3y² — I

n!
ar = — (n — r)!^'

(i) y ; (ii) 2y — y .
36.

41.

52. 53.

30.
71.

73. 75.

ANSWERS TO THE EXAMPLES    3⁸9 5
9. x—^x +^{x —x}....    10. -+^x+^x5-~ ....    12. -4_b.
3    5    7    4 2 4    22 3
13.    ⁺3a    81a3....    14. x^—_ix³+aix⁵....    17. -4k.
18. 1 + i x + ~^Lx2+ ] Ax³ ....    20.    ^I= - + x3 - ^x5e^x + 1 2 4 48 480^....22. ^px–s^p(p²+6)^x3+,:^',-,^p(^p4+2^op2+^6o)^x5....

Examples 12.

1. Max. 1; Min. — i i .    2. Max. 9a³ ; Min. — 3
3. Max. 3 ; Min. A.    4. Max. r, 2 ; ; Min. o.
5. Max. 73; Min. 69, 69.    6. Max. 6₃,6₃; Min. —

7.Max. 13; Min. — lo; Point of inflexion where x = o.
8.Max. 2 V ab + a + b _min.2 V ab — a — b9. Max. ,₇4_T; Min. o.

2'lab—a—b

'lab+a+b

10. Max. —

'dab;

Min.

Va.

11.

12.

Max.

41;

Min.

3.13.

zo7

14. e

e .

16.

3—3log

17.

a=—6;

$=9.

125 yards from A along AB and then across the grass to C.

20. n parts.21. * {(a + b) — Va2 — ab + b²}.

₆

(a* +

b3)2

Max.

'lab +

;

—

'ab

—

V~2 -

32.

165

feet.

33.

y =

34.

Max. z ; Min. .

35.

Max. —

Min.

36.

Max.

12;

Min.

8, — zoo.

38. x = o.    39. ± --ⁱ=.    40. ±
J₂1~3
43. Max. R; Min. o.    44. Max. 4^.317; Min. 4^.183.
45. Max. V; Min.    46. — 5.    47. (a + b)².
22. 24.

27.

30. 2 'lab.    31. _f(a + b)².
23. 66 minutes.

390        ACTUARIAL MATHEMATICS
Examples 13.
1. o.    2. o.    3. 1.    4.

log, a.

7. — I.    8.    /?nma3(m—n).
11. 1.    12.
9I
m    ₆_{_}I
ⁿV za
10. r.
13. log_e a.    14. I re    15. I.    16. a',. 24

17. s, 18. ^b+alog^b. 20. oo. 2b^—a a

21. a = 120; b = 6o; c = 180.    23. o.    24. o.
2
25. 3U.    ' 26. _}u.    28. — x      + y    30. u — uy.
y² + x
33. .02455; — '0003; 0.    34. • 109.    36. keax+Ie^:_,
Examples 14.
Note. In the answers to questions on indefinite integrals, the presence of the constant of integration is to be inferred.
_2.-ⁿ⁺¹.
1_e2x
n+1' 3(—n+I)'

3.ix⁵ + 3x³a² + xa⁴; ax + bx² + 3cx³; — } (I + x)^—2.

4.— cosx; sinx; — cotx.5. log_e a^;log_e a + bx; log_e a + hbx² + cx.

6.— i cos 2x; I sin 3x; i tan 4X.

7.l01 . — _I

(

I); 1 –

12 '

(a +

3x)

^4'

8.^31ogx+x'(n— ^3a) x^n-2+(n— 2)xⁿ ²⁺(n
9.ilogx₊I; logx — tan' x.
10.a sin^—1 x; _asin ¹ax.11.sin 3x + - sin x.

— (n — 3) ^yn_3(n — 2) y^n—2+ (n — ^I) ^yn—1J^'

L(n –

3) (

ⁿ

(n –

^n-2

(n –

I) (I

x)n-l] .

l l

^{+m y}

14.

dxx2

15.

log f

''Vx

—

2xl

t x

^–

I r

38.

48;

24.

ANSWERS TO THE EXAMPLES    39¹
1+x
16. – log (I – x) – tan' x.    17. A log    - -
I – x 2(I+x)
23. e^x+ax⁰+bx+c.    25. IIa
₆-_*

26. 1x = ke^—Ax^—BeIloge c 27. A log I + t. 28. log _I^I_x4^.Examples 15.

Note. In the answers to questions on indefinite integrals, the presence of the constant of integration is to be inferred.
I    I    2    I    I    I
1' n — I (I + x)^n—' + n — 2 (I + x)^R—2n - 3 . (I + x)n—3' 2 ~x + - ,
(x— I)(x+2)ⁱ.
2. 2 log (x - 3) - log (x - I); 3X + I I log (x - 2) - 2 log (x - I); 2Vx—1 3    2
35    (5^x + 6x + 8x + 16).
. 1 10x0-a0' —    + 1^'x3
^gx^z+a²'    x -    ag
4az    x₊₃

4.} log (3 + 4 sin x) ; log (4x⁰ + 3) ; log (e^x + cos x).

5.sin-' 2x +log (x++ ^V2+x+x²).

6.– 3 log {A + } cos x}; log I (+ x +) xz + V- tan 1 2x+ 1

1-x

⁰

- a

log \

x-+x+

^31og

V1 + x²; _jlog (4x⁰ + 3) + -2 - tan +' ²¹

V3V3

A {cos x - k cos 5x} ; sin Ax + + sin

n —

3 (x2 - 2a²) 'c_t⁰ + x²; ^Ilog - I -± x¹

1 +

ⁿ

cx+b

/2+x

_zklogcx+b+kor _ktan-¹ _k12. -2 \/2

(i) Vx² - 1 ; (ii) {x 1/xz - + log (x + Vx' - I));

(iii)

—

+ Vx2 — I.

— 4 log (x — I) — 2 (x ⁵- 1) — 2 (x^I 1)2 tan-^' x + I log (I + x²).

log t; t; log ^t_t2.16. ^Ax'log x – ix2; _a3(2 – 2ax + a²x²).

17. e

—

2x + 2);

log x —

392 ACTUARIAL MATHEMATICS

sin x — x cos x ; x' sin x + 3x² cos x — 6x sin x — 6 cos x.

tan'

x — x}; e

/(x

I).

^eax

cos

sin

bx)/(a

-i log (2x + V4x² — 7).22. log tan (far + ix).

23. — {

log

—

x)}

24.

log

x {

log (log

x) —

I}.

25,

(3a

2x2)

₂₆

zbx +

c)"`}

4 (a' +

x2)T2 (il

+ I)

~_{log (6x + I) + 2 V3 ^/3x² + x + 8}.

x — log_e (I + c^x)/loa_e c.

log

Vx2

+ I

+ _ tan

^-I {

(2x +

I)/

V3}.

V5 + zx + x² — log (I + x + V₅+ zx + x²).

log x — } log (I + x²).32.tan^-I (k tan ix).

33.

log

x—'~2+VI+x

34. —

sin

^i3—X

V2 x + V2 ++ x

35. —

log{x+4+2'

2+x—

²}

₁

log x.

N'2

sec

⁼

36.

log

37.

sin

^-I

(x/a) —

— x

2(a—b)a + b tanx

38.

Q {

ⁱ

— x

³}

39. .?_, {

3x — 4 log

cos

+ 3 sin x)}.
3a

( V

40.

tan-

tan x}

+ab

IVa+b

(^a) ^y(^{log y}— ^I) ; (b) ^ycosy — V I — y².

— I (2x² + I)/(x² + I)² + a constant.45. ?} tan^s x².

46.

log (x +

Vx2 —

I) —

sec

^—I

47.

_a2

_b2

tan-I {

—+

tan

ix}.

48. —

+ x2)- (2a

+ 3x

49.

(x/q — I/q2).

3—2m

(2 — 2m)

— a

)

^m—I

(2 — zm) a=

tn—1

x—a

—

a2 4a3 log

51.

^ix4

(log x)

—

⁴

log

+ 3

_ x

⁴

— 2 {a (a — x) 1^/tax — x' + a' sin-^I (a x)/a} — (tax — x2)!.

sin (x — a)

sin (a — b) ^logsin (x — b)

50.

ANSWERS TO THE EXAMPLES 393
55. x³ sin ¹x + (2 + x²) I - x². 56. tan^-1 (- cos 20).

^Itan-¹ /3=x-(5-2x) V(x-2)(3-x)}.

ke-""xpe',

where

p =

B/log

log

x — log {(x

V3x

2x + r}.

61,

tan_

_= tan x - x.

V2V2

1/x4 + x

—

₄

_(x2

1)2'

63.

₁\₃z^-¹ - izi -where z (2 - x³)/x³.

*(2+x^- )T-1-(2+x-1)1.

6 ( z' - _/z⁸+ 4.e - Sz^e + z⁵- z⁴), where z (x + I)k.

'a-b(t+ ^I I.

^t3

a4n -(z - 4b log z - 6b²z^-1+ b³ z ²- lb4 ^z-3),

where

cos x

60.

logtanix

Sin

62.

^log{(^x4— ^I)²/(^x8+ x⁴+ I)} +
4 '\^/3

tan-¹ {(2x⁴ + 1)/ V₃}.
70.

71. ^I-. cos_ I + 2 cos B
1 _ _
V₃ 2+cos0

Examples 16.

nⁿ-^E1/(n + I); I2a + 41;b.
(I — _e—a¹k)_/a2_; 2 [51(m+n+2) — (— 5)i (m+n+21/(m + n + 2).
4. I - ^I= ; i 7, - log 2.    5. log ; 2 - f log + 1r/6 .¹3. ^/-2
6. 17^log2+4z; 7,/V3.    7.    1/a(''2—I); V2^—I.
8. 37,/16; 77/15.    9. l ; rsoh•

10.(b² log b - a² log a) - (^b2- ^a2) ;(^e2+ ^{e 2}).
11.^IT-2; -26e- +16.13.log 1.

15. }

(b-a

^—

(3b -

3a-I)

^—

(a-b)'-

log (b - a).

16.

₃

7,/16.

18. -

log 2 -

19.

log

24,

log

(3) -

26.

+ N

log

⁵

+ tan

^-1

28.

36a

29.

3a2 (c

30.; .

31.

_8a2/3.

32.

?d-

33. -

(

⁹

^-9

34.

V3/5.

35.

I -

log

36.

2. 3.

(

^87,

3 V3)

^a2;

(47

3) a2•

37.

47,.

ACTUARIAL MATHEMATICS

40. ₃(z'~ — I) — log 2; 1 (2ⁱ + I) + log 2. I    ₄₂n (a — Sin a COS a)
+ Vac' +a2)    sin'a
44. n!/aⁿ⁺¹; a log (a + I).    45. y = 4a sin^-1 V2x/a + './zax - 4x⁸.
47. 3 V₍2/8 + log (V2 + I).    l

49. x⁸ { -₂+ log (I + 2)₎ ' 3 {-^I_ +log (1 + V2)1. 50. ia².
Examples 17.

2. 9921. 3. zo•4 miles.

8.(95^u0 - 5^Ou1 + 600u₂ - 350u_% + 425u₄).
9.zoo square inches.11. •8571...; (i) •8806...; (ii) •8946....

12.

The second.

13.

29,426.

14.

888.

16.

•6942.

22.

•5236.

24.

3 142....

25. 1'1358....

26.

6931....

27.

142....

Examples 18.

^4.

_32.

_ii.

^7.

as•

y.10. (2n)

(

₂

1)2n

11.

⁵

₁

n! n! \

12.

_WI5.

13.

14.

15.

₅

16.

¹⁵

₆

³⁵

17. (a)

₂

_(b)

₂

(c)

^(m2

)

18.$°i.19.

20.

⁷

21.

_Tiiu.

22.

138

^{a 56}

23. A .
5

24.

(I)

Iaa; (2)

26. i.27.

u.28.

29.

18.

31.

_(T73)2.

32.

AR ;

33.

34.

MA-

^36.36.

^37.

NA.

38.

(i)

^p2

(

^zq

)

(ii)

p(p 4)

i'T;

1i;

(p+q)(p2+pq+q2)'

+pq+q

40.

Hi.

41.42.

&I.

43.

gR..

44.

6s. od.;

5s.

od.

45.

35s.

od.

46. 6:1.48.

Purse

49.

(i) 63s.

od.; (ii)

56s.

od.

50.

₁

⁷

;

8i.

51.

1).

52. io.53.

gym.

(1) --n; (2) +.1g; (3) g: Most probable number is 4; =13^.

i26.56. 2^.06 x to ~.57. (I) k; (2) Ws; (3) 4.

58.

•1o5.

60.

•98304 or •09888.

61.

(1) •72;

(2)

•504.

62. (I)(2)

•000007..., '001 ...

63.

(a)

972;

(b)

•271.

394

39.

16a

/3.

41.\

395
6911 + 3 ₂m+'^.
ANSWERS TO THE EXAMPLES 64. HI.    66. I —    n.    67. I 1 ; 6s. id.    68. 7.
74. (^I) A; (²) Tt o; (3) JLOb'    75. e^' g4.
76. (I) I shilling; (2) 141 shillings; (3) It shillings.

z
77. 3n — 3ⁿ + z    78.
9n^z — 9n + 2

80. (I) agi ; (²) _ ; (3) 44'

log_e z
loge 2.

loge (I —
81. 14A; Fo⁵⁰³3^'a
^'520    '
79. _sue. 82. 3003
₂15 '
    3ⁿ
84. 36n^z — 18n + 2    86, h a .
88. A.    89. '506....
83. (I) l+ ; (2)    ; (3) 1^.87. ⁽⁴¹⁾¹³^{(3 !)4},
₅₂1
90. (1) Between 5 and 6; (2) 4; (3) 7.    ^92.ii,⁹s•
93. (^I) :38,; (²) y; (3) +4.    94. Vi_i.    95. int.
96. 3000: 3007.    97. I.    ^98._1.100. A.    101, A.
102. Ha.    103. (I) 1; (²) {; (3) ibo'    104. -₁^q.
105. (a)    ; (^b) 15; (^c) A.    ^107._Zip•^108.20822'

Examples 19.

z
2.    (n + 2).    3. _az4, n loge 2.    5. 61 (~2    I).
(I) ^?z; (2) ^Y9. 4
7. - a^z; '_.    8.

77^,    4 11, lay.    12. ha3.    13.
3
8 log 2 - 3 64 log 2 - 15

14. A.    15. H.    16.
c^z 17.1.    18. g.    19. a)^.
20. 1 — 1 ^loge 3.    21. 4 .    22. 1.    24.    + 1 log_e 2.
25. Al; _)dmiles.        26. A (2 — ''2) + log (I + V2). Miscellaneous Examples.
1 . I 2 (n³ - 4n^z + I In + 4).    2. H.    3. Max. 1 ; Min. - 1; Area A.

5. (I) 3 (1    xz) (x + a_x) - ; _(ii)V (I — x^z){I^I — (sin-1 x)zi

_6.₍₁₎I ; 01)1.^7.7 1 .I 8. x - log (x^z + 2x + 3) + 2 tari-¹ ` ~a , a^z log tan (}a + )x) + b^zsin x.

396    ACTUARIAL MATHEMATICS
9. (a) ¹9,7⁶4; (^b) 59,¹5^6. 2
12.    ^1    I-
+ x²' I + x^2'I + x²• 19. A=18,y=2^.8.
26. 977,961; 1023.
11. (a) 293; (b) 152; (c) 141.
14. 5'225.    16. (^I) sa; (²) a•

23. _0577...ft. per min.

m!n! m!(n

1)! (m+n)!' (m+n)!

21. ,4.

27.    2b
3(^b-^c)^.
28.
29. — 1.    30. o294; •o492; •0682.
31. na° 16 •
32. * ft. per min.    33. 14 hr. 41 min.    36. — 1^.88.
40. ?--a + !-a log_e 3.
42. 7T
+ i log 2 — *;
12    16^'
48. 86; 121.    49. ;.    50. 4e^—1.    51. 48^.68... miles.
56. _I- 31
II⁴

57.(i) x — log (I + e^x); (ii) log (I + e^x); (iii) ex — log (I + C^x); (iv) log (I + e^x) — x — e^x.

x _

3a²81a⁸

61..62.63.

_t2{

^t+1

(log

I) —

1}.

65.

4°9

ft. per

sec.66.

6°;

⁷

6°.

67.

₂

—

68.

113

6, 117

120

4, 124

2, 128

34'

'4,

49'4,

⁸

⁶

168

6, 179

70.

• 14 or • 18.

71. (i)

(zx — 3)

+ x) ; (ii) ,';. L7x — 23 log

sin

+ cos x)].

75.

2'4695....76. 28•o....

78.

^aH

80.

(i) v

(ii) a.

82.

4 {l +

(log x)

—

log

x};

* sin

x —

* sin

(i) 2 V'-X- - 2 V'k log h ^/k + .Vx}; (ii) _3a{x - (x - a)i};

(iii) 2

'V -

tan

^—1

v'x.

; 1.90. 10^.476.

sin

x log+

—

95.

1403,1498,1598,1703,1814,1931,2054,2182,2316,2458,2609;120.

98.

[x (x -

1) +

c] [(x -

-2)

c] ...

[2.1 + c], where

and c are arbitrary constants.

100.

030....

101.".

102.

2 {212+2 +

₃

12+1 - 712

- n - 7).

38. —

12.

39.

I —

(A5)6

(ia)

⁶

3~~3

44. 5

⁸

'93

^21....

52. 0'109....55. 7'04....

59.

^II-{

⁴

- 212

312

2214.

60.

5'975

81. 29

08.

92.

91.

N!N'

93. 15 :

17.