Question

In a geometric series with first term, a and common ratio, r (where r is real and 1r), the sum of the first 7 terms is 4 times the sum of the following 7 terms. Find the ratio of the sum of the first 21 terms to the sum of the first 14 terms.

Answer 1

Answer:

उकेतूतूएम573कक57247ज1257ल741कुक456ज37के4ककु5स75लवल8518

Step-by-step explanation:

कु4के4कयक64कक6क3ई64की63इक़्क़464व6ई3क8ऊ8व7725o5359573ओ64ई2

Answer 2

Given:

A geometric series

Sum of first 7 terms, $S_{7}=4(A_{8}+...+A_{14})$

To find:

Ratio of the sum of the first 21 terms to the sum of the first 14 terms.

Solution:

A geometric series is given by: $a,ar,ar^{2},ar^{3}...ar^{n}$ where $a$ is the first term and $r$ is the common ratio. The sum of its $n$ terms is given by the formula:$S_{n}=\frac{a(r^{n}-1)}{(r-1)}$

Here, the sum of first 7 terms is $S_{7}$ with $n=7$ and the sum of the next 7 terms is given by $A_{8}+A_{9}+...+A_{14}$

$S_{7}=\frac{a(r^{7}-1)}{r-1}=4(A_{8}+A_{9}+...A_{14})$

$A_{1}+A_{2}+...+A_{7}=4(A_{8}+A_{9}+...+A_{14})$

$a+ar+ar^{2}+...+ar^{6}=4(ar^{7}+ar^{8}+...+ar^{13})$

$a(1+r+r^{2}+r^{3}+r^{4}+r^{5}+r^{6})=4ar^{7}(1+r+r^{2}+r^{3}+r^{4}+r^{5}+r^{6})$

$r^{7}=\frac{1}{4}$

Sum of first 14 terms, $S_{14}=\frac{a(r^{14}-1)}{(r-1)}$

Sum of first 21 terms, $S_{21}=\frac{a(r^{21}-1)}{(r-1)}$

Ratio of sum of first 21 terms to sum of first 14 terms is given by:

$\frac{S_{21}}{S_{14}}=\frac{a(r^{21}-1)}{(r-1)}*\frac{(r-1)}{a(r^{14}-1)}$

$\frac{S_{21}}{S_{14}}=\frac{r^{21}-1}{r^{14}-1}$

$r^{21}=(r^{7})^{3}=(\frac{1}{4}) ^{3}=\frac{1}{64}$

$r^{14}=(r^{7})^{2}=(\frac{1}{4})^{2} =\frac{1}{16}$

Substituting these values in the equation,

$\frac{S_{21}}{S_{14}}=\frac{\frac{1}{64}-1 }{\frac{1}{16}-1 }$

$\frac{S_{21}}{S_{14}}=\frac{1-64}{64}*\frac{16}{1-16}$

$\frac{S_{21}}{S_{14}} =\frac{21}{20}$

Thus, the ratio of the sum of the first 21 terms to the sum of the first 14 terms is$S_{21}:S_{14}=21:20$ .

The ratio of the sum of the first 21 terms to the sum of the first 14 terms in a geometric series with the first term 'a' and common ratio 'r', is 21 : 20.