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arlik [135]
2 years ago
9

(b) If a series follows geometric progression, show that the ratio of the sum of term to the sum from (n+1) term to (2n) term is

4 marks]
Mathematics
1 answer:
Nadusha1986 [10]2 years ago
5 0

Proof with explanation:

We know that the sum of first 'n' terms of a Geometric progression is given by

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

where

a = first term of G.P

r is the common ratio

'n' is the number of terms

 Thus the sum of 'n' terms is

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

Now the sum of first '2n' terms is

S_{2n}=\frac{a(1-r^{2n})}{1-r}

Now the sum of terms from (n+1)^{th} to (2n)^{th} term is S_{2n}-S_{n}

Thus the ratio becomes

\frac{S_{n}}{S_{2n}-S_{n}}\\\\=\frac{\frac{a(1-r^{n})}{1-r}}{\frac{a(1-r^{2n})}{1-r}-\frac{a(1-r^{n})}{1-r}}\\\\=\frac{1-r^{n}}{r^{n}-r^{2n}}\\\\=\frac{1-r^{n}}{r^{n}(1-r^{n})}\\\\=\frac{1}{r^{n}}

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