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Find the sum ...

Question

Find the sum of the following geometric series:
$0.15 + 0.015 + 0.0015 + . . . .$ to 8 terms

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Solution

$a = 0.15$

$r = \frac{0.015}{0.15} = \frac{1}{10}$

$S_{8} = \frac{a (1 - r^{8})}{1 - r}$

$= \frac{(0.15) (1 - {(\frac{1}{10})}^{8})}{(1 - \frac{1}{10})}$

$= \frac{(0.15) (0.99999999) \times 10}{9}$

$= 0.166666665$ .

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Find the sum of the following geometric series :

(i) 0.15 + 0.015 + 0.0015 + ..... to 8 terms ;

(ii) $\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2 \sqrt{2}} + . . . . .$ to 8 terms ;

(iii) $\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . . .$ to 5 terms.

(iv) $(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y + y^{3}) + . . . .$ to n terms ;

(v) $\frac{3}{5} + \frac{4}{5^{2}} + \frac{3}{5^{3}} + \frac{4}{5^{4}} + . . . . .$ to 2n terms ;

(vi) $\frac{a}{1 + i} + \frac{a}{(1 + i)^{2}} + \frac{a}{(1 + i)^{3}} + . . . . + \frac{a}{(1 + i)^{n}}$

(vii) $1, - a, a^{2}, - a^{3}, . . . . . . . .$ to n terms $(a \neq 1)$

(viii) $x^{3}, x^{5}, x^{7}, . . . .$ to n terms

(ix) $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, . . . .$ to n terms

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