Question

the ratio of the 4th and 12 term of a geometric progression with a positive common ratio is 1 : 256 if the difference of the two terms be 61.2 find the sum of 8 terms of the series.que no.19

Answer 1

Answer:

Sum of the 8 terms of the G.P is 7.65

Step-by-step explanation:

Formula used:

The n th term of the G.P a, ar, ar²........... is

$t_n=a\:r^{n-1}$

The sum of n terms of the G.P a,ar,ar²....... is

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

Given:

$t_4:t_{12}=1:256$

$\frac{t_4}{t_{12}}=\frac{1}{256}$

$\frac{a\:r^3}{a\:r^{11}}=\frac{1}{256}$

$\frac{1}{r^8}=\frac{1}{256}$

$r^8=256$

$r^8=2^8$

$r=2$

Also,

$t_{12}-t_4=61.2$

$a\:r^{11}-a\:r^3=61.2$

$a(r^{11}-r^3)=61.2$

$a(2^{11}-2^3)=61.2$

$a(2048-8)=61.2$

$a(2040)=61.2$

$a=\frac{61.2}{2040}$

$a=\frac{612}{20400}$

$a=\frac{3}{100}$

Now,

Sum of the 8 terms of the G.P

$=S_8$

$=\frac{a(r^8-1)}{r-1}$

$=\frac{\frac{3}{100}(2^8-1)}{2-1}$

$=\frac{\frac{3}{100}(256-1)}{1}$

$=\frac{3}{100}(255)$

$=\frac{3}{20}(51)$

$=\frac{153}{20}$

$=7.65$