First term of an infinite g.p. is 4 and sum is 8. Find Common ratio.
The Correct Answer and Explanation is:
To find the common ratio of an infinite geometric progression (G.P.), we can use the formula for the sum of an infinite G.P.: S∞=a1−rS_{\infty} = \frac{a}{1 – r}S∞=1−ra
Where:
- S∞S_{\infty}S∞ is the sum of the infinite G.P.
- aaa is the first term.
- rrr is the common ratio.
We are given the following information:
- First term a=4a = 4a=4.
- Sum of the series S∞=8S_{\infty} = 8S∞=8.
Substituting the given values into the formula: 8=41−r8 = \frac{4}{1 – r}8=1−r4
Now, solve for rrr:
- Multiply both sides by (1−r)(1 – r)(1−r):
8(1−r)=48(1 – r) = 48(1−r)=4
- Expand the left-hand side:
8−8r=48 – 8r = 48−8r=4
- Subtract 8 from both sides:
−8r=−4-8r = -4−8r=−4
- Divide by -8:
r=−4−8=12r = \frac{-4}{-8} = \frac{1}{2}r=−8−4=21
So, the common ratio rrr is 12\frac{1}{2}21.
Explanation:
In a geometric progression, the sum of the infinite terms exists only when the absolute value of the common ratio ∣r∣<1|r| < 1∣r∣<1. This is why the sum converges to a finite value. The formula S∞=a1−rS_{\infty} = \frac{a}{1 – r}S∞=1−ra applies to such series, where aaa is the first term and rrr is the common ratio. Given that the sum is finite (8), we could solve for the common ratio by rearranging the formula, resulting in r=12r = \frac{1}{2}r=21.
