Question

What is the sum of the first nine terms of the geometric sequence 20, 10, 5, ...?

Answer 1

SOLUTION

TO DETERMINE

The sum of the first nine terms of the geometric sequence 20, 10, 5, ...

EVALUATION

Here the given geometric sequence is

20, 10, 5, ...

First term = a = 20

Common Ratio = r = 10/20 = 1/2 ( < 1 )

Number of terms = n = 9

Hence the required sum of first nine terms

$\displaystyle \sf{ = 20 \times \frac{1 - { \bigg( \frac{1}{2} \bigg)}^{9} }{1 - \frac{1}{2} } }$

$\displaystyle \sf{ = 20 \times \frac{ \frac{ {2}^{9} - 1 }{ {2}^{9} } }{ \frac{1}{2} } }$

$\displaystyle \sf{ = 20 \times \frac{ {2}^{9} - 1 }{ {2}^{8} }}$

$\displaystyle \sf{ = 20 \times \frac{ 512 - 1 }{ 256}}$

$\displaystyle \sf{ = 20 \times \frac{ 511 }{ 256}}$

$\displaystyle \sf{ = \frac{ 2555 }{64}}$

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