6. When 3 per cent. is the rate of interest, find what sum must be paid now to receive a freehold estate of £320 a year 10 years hence; having given

log 1.0320136797,

log 7.297988632030.

7. Supposing an annuity to continue for ever to be worth 25 years' purchase, find the annuity to continue for 3 years which can be purchased for £625.

8. A sum of £1000 is lent to be repaid with interest at 4 per cent. by annual instalments, beginning with £40 at the end of the first year, and increasing 30 per cent. each year on the last preceding instalment. Find when the debt will be paid off; having given

9. What is the present value of an annuity which is to commence at the end of p years, and to continue for ever, each payment being m times the preceding? What limitation is there as to m?

10. What sum will amount to £1 in 20 years, at 5 per cent., the interest being supposed to be payable every instant?

th

11. If interest be payable every instant, and the interest for one year be of the principal, find the amount in n years.

12. A person borrows a sum of money, and pays off at the end of each year as much of the principal as he pays interest for that year; find how much he owes at the end of n years.

13. An estate, the clear annual value of which is £A is let on a lease of 20 years, renewable every 7 years on payment of a fine; calculate the fine to be paid on renewing, interest being allowed at six per cent.; having given

log 1062.0253059,

log 4.688385.6710233,

log 3.1180424938820.

14. A person with a capital of £a, for which he receives interest at r per cent., spends every year £b, which is more than his original income. In how many years will he be ruined?

Ex. If a = = 1000, r=5, b=90, shew that he will be ruined before the end of the 17th year; having given

We shall confine our attention to continued fractions of the

For the sake of abbreviation the continued fraction is some

......

When the number of the terms a, b, c, is finite, the continued fraction is said to be terminating; such a continued fraction may be reduced to an ordinary fraction by effecting the operations indicated.

601. To convert any given fraction into a continued fraction.

m

n

Let be the given fraction; divide m by n, let a be the quotient and p the remainder; thus

divide n by p, let b be the quotient and q the remainder; thus

If m be less than n, the first quotient a is zero.

We see then that to convert a given fraction into a continued fraction, we have to proceed as if we were finding the greatest common measure of the numerator and denominator, and we must therefore at last arrive at a point where the remainder is zero and the operation terminates; hence every fraction can be converted into a terminating continued fraction.

602. The fractions formed by taking one, two, three, &c. of

1

1

b+c+ &c.

the quotients of the continued fraction a + are called converging fractions or convergents. Thus the first convergent is a;

1

ab + 1

the second is formed from a +

it is therefore

; the third

b

603.

The convergents taken in order are alternately less and

greater than the continued fraction.

The first convergent a is too small because the part

1

b+ &c.

1

omitted; a + b

is too great because the denominator b is too

604. To prove the law of formation of the successive con

vergents.

a

ab + 1 abc+a+c
I' b
bc + 1

The first three convergents are ; the numerator of the third is c (ab + 1) + a, that is, it may be formed by multiplying the numerator of the second by the third quotient, and adding the numerator of the first; the denominator of the third fraction may be formed in a similar manner by multiplying the denominator of the second by the third quotient, and adding the denominator of the first. We shall now shew by induction that such a law holds universally.

p p' p"

Let Pp'

be three consecutive convergents, m, m', m",

the corresponding quotients; and suppose that

Let m" be the next quotient; then the next convergent

only in taking in the additional quotient m"",

If therefore we suppose

p" =m'"p" + p′ and q′′ = m′′q′′ +q',

"

the next convergent to will be equal to

تو

be formed by the same law that was supposed to

p"

hold for ; but the law has been proved to be applicable for

the third convergent, and therefore it is applicable for every subsequent convergent.

We have thus shewn that the successive convergents may be formed according to a certain law; as yet we have not proved that when they are so formed each convergent is in its lowest terms, but this will be proved in Art. 606.

605. The difference between any two consecutive convergents is a fraction whose numerator is unity, and denominator the product of the denominators of the convergents.

This is obvious with respect to the first and second convergents, for

Ρ

ab + 1
b

α

1

1

Suppose the law to hold for any two consecutive convergents

; that is, suppose p'q − pq' =±1, so that

then, p"q-p'q" = (m"p' + p) q -p' (m"q' + q) = pq' — qp' = 1,

thus the law holds for the next convergent. Hence it is universally true.

606. All convergents are in their lowest terms.

For if the numerator and denominator of had any common q

measure it would divide p'q- pq' or unity, which is impossible.

607. Every convergent is nearer to the continued fraction than any of the preceding convergents.

We shall prove this by shewing that every convergent is nearer to the continued fraction than the preceding convergent.