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Find common ratio when sum of n terms of geometric progression is givenIn this section you will learn to find common ratio when sum of n terms of geometric progression is given.Here we will use the following formulas : 1) $a_{n} = ar^{n  1}$ 2) $S_{n} = a_{1}\left ( \frac{r^{n}  1}{r  1} \right )$ where a = first term r = common ratio n = number of terms Examples on find common ratio when sum of n terms of geometric progression is given1) Determine the common ratio and the number of terms in G.P. If $a_{1}$ = 3, $a_{n}$ = 96 and $S_{n} $= 189Solution : Let 'r' be the common ratio of the given G.P. then $a_{n} = 96 ⇒ a_{1} r^{n  1}$ = 96 ⇒ 3$r^{n  1}$ = 96 $r^{n  1}$ = 32 (1) Now, $S_{n} $= 189 $S_{n} = a_{1}\left ( \frac{r^{n}  1}{r  1} \right )$ $a_{1}\left ( \frac{r^{n}  1}{r  1} \right )$ = 189 $r^{n} = r^{n 1} \times$ r So, 3$\left ( \frac{(r^{n1})r  1}{r  1} \right )$ = 189 but $r^{n 1}$= 32 and $a_{1}$ = 3 ∴ 3$\left ( \frac{32r  1}{r  1} \right )$ = 189 $\left ( \frac{32r  1}{r  1} \right )$ = 63 32r 1 = 63(r 1) 32r  1 = 63r 63 32r  63r = 63 + 1 31r = 62 ∴ r = 2 Put r = 2 in equation (1) we get, $2^{n  1}$ = 32 $2^{n  1} = 2^{5}$ ∴ n  1 = 5 ⇒ n = 6 So, common ratio is 2 and number of terms are 6. 2) Find the common ratio of G.P., if first term is 2, last term is 486 and the sum of these terms be 728. Solution : Let 'r' be the common ratio of the given G.P. then $a_{n} = 486 ⇒ a_{1} r^{n  1}$ = 486 ⇒ 2$r^{n  1}$ = 486 $r^{n  1}$ = 243 (1) Now, $S_{n} $= 728 $S_{n} = a_{1}\left ( \frac{r^{n}  1}{r  1} \right )$ $a_{1}\left ( \frac{r^{n}  1}{r  1} \right )$ = 728 $r^{n} = r^{n 1} \times$ r So, $a_{1}\left ( \frac{(r^{n1})r  1}{r  1} \right )$ = 728 but $r^{n 1}$= 243 and $a_{1}$ = 2 2$\left ( \frac{(243r  1}{r  1} \right )$ = 728 $\left ( \frac{(243r  1}{r  1} \right )$ = 364 364r 364 = 243r 1 364r  243r = 1 + 364 121r = 363 ∴ r = 3 So common ratio is 3 From find common ratio when sum of n terms of geometric progression is given to Home Covid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! Covid19 has affected physical interactions between people. Don't let it affect your learning.
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