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 ly receive the same improvement during the term, as if it had been laid up at interest at its commencement.

126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in $$n$$ years, the number of periods of conversion will be $$\nu n$$, in the

1st Case, Where the interest is convertible $$\mu$$ times in each of the intervals between the payments of the annuity, we have $$\left ( 1+\frac{r}{\nu} \right ) ^{\nu n} \text{V} = \frac{a}{\pi} \cdot \frac{\left ( 1+\frac{r}{\nu} \right ) ^{\nu n} - 1}{\left ( 1+\frac{r}{\nu} \right ) ^\mu - 1} = \text{M}$$, (117, 121, and 125). In the 2d Case, when the annuity is payable $$\mu$$ times, in each interval between the conversions of interest, $$\left ( 1+\frac{r}{\nu} \right ) ^{\nu n} \text{V} = a \left [ \frac{1}{r}+\frac{\mu-1}{2\pi} \right ] \cdot \left [ (1+\frac{r}{\nu})^{\nu n} - 1 \right ] = \text{M}$$, (117, 122, and 125).

And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible, $$\left ( 1+\frac{r}{\nu} \right ) ^{\nu n} \text{V} = \frac{a}{r} \left [ (1+\frac{r}{\nu})^{\nu n} - 1 \right ] = \text{M}$$, (117, 123, and 125).

127. According as $$\nu$$ and $$\pi$$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $$\text{M}$$ be denoted by $$y'$$, $$h'$$, $$q'$$, or $$c'$$;

128. Example 1. What will L. 320 amount to, when improved at compound interest during 40 years; the rate of interest being 4 ''per cent. per annum?''

By the first formula in No. 116, the operation will be as follows:

And the answer is L. 1536, 6s. 6½d.

129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:

So that in this case the amount would be L. 1560, 2s. 9½d.

130. Ex. 3. Required the present value of an annuity of L. 250 for 30 years, reckoning interest at 5 per cent.

By the third formula in No. 116, the operation will be thus:

And the required value is L. 3843, 2s, 3¼d.

131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122, $$\nu$$ will be equal to 1, $$\pi=2$$, and $$\mu=2$$; that formula will therefore become $$a \left ( \frac{1}{r}+\frac{1}{4} \right ) \cdot \left [1-(1+r)^{-n} \right ] = \text{V}$$; and the operation will be thus:

The value of the annuity will, therefore, in this case, be L. 3891, 3s.

132. Ex. 5. To what sum will an annuity of L. 120 for 20 years amount, when each payment is improved at compound interest, from the time of its becoming due until the expiration of the term; the rate of interest being 6 per cent.?

The operation by the second formula in No. 116 is thus:

And the amount required is L. 4414, 5s, 5d.

133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulæ given in No. 127; which, in this case, will become $$\frac{120}{0{\cdot}6} \left [ (1{\cdot}03)^{40} - 1 \right ]$$, and the operation will be as follows: