Question

for the G.P, the ratio of sum of the fifth and third to the difference of the fifth and third term is 5:3.find r​

Answer 1

Answer:

Step-by-step explanation:

Hard

Answer 2

The value of r is 2.

Step-by-step explanation:

Let  a = 1st term of a G.P

r = common ratio of a G.P

$a_{n}$ = nth term of a G.P = a$r^{n-1}$

Third term = $a_{3}$ = a$r^{3-1}$ = $ar^{2}$

Fifth term = $a_{5}$ = $ar^{5-1} = ar^{4}$

Sum of fifth and third term = $ar^{2} +ar^{4} = ar^{2} (1+r^{2} )$

Difference of fifth and third term = $ar^{4} -ar^{2} = ar^{2} (r^{2} -1)$

Ratio of sum of fifth and third to the difference of fifth and third term = 5 : 3

=> $\frac{ar^{2}(1+r^{2} )}{ar^{2} (r^{2} -1)}$ = $\frac{5}{3}$

=> $\frac{(1+r^{2}) }{(r^{2} -1)} = \frac{5}{3}$

=> $3 + 3r^{2} = 5r^{2} - 5$

=> $5r^{2} - 3r^{2} = 3 + 5 = 8$

$=> 2r^{2} = 8\\=> r^{2} = \frac{8}{2} = 4\\=> r = \sqrt{4} = 2$

Hence, the value of r is 2.