Consider an infinite geometric series with first term and common ratio . If its sum is 4 and the second term is , then [2000]

Question

Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is 4 and the second term is $\frac{3}{4}$ , then [2000]

Smart Achievers · Accepted Answer

(4)

$Sum = 4 and second term = \frac{3}{4}$

$\Rightarrow \frac{a}{1 - r} = 4 and a r = \frac{3}{4} \Rightarrow r = \frac{3}{4 a}$

$∴$ $\frac{a}{1 - \frac{3}{4 a}} = 4 \Rightarrow \frac{4 a^{2}}{4 a - 3} = 4$

$\Rightarrow a^{2} - 4 a + 3 = 0$ $∴$ $a = 1 or 3$

$When a = 1, r = \frac{3}{4} and when a = 3, r = \frac{1}{4}$

Solution

Q.
Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is 4 and the second term is $\frac{3}{4}$ , then [2000]

1 $a = \frac{4}{7}, r = \frac{3}{7}$

2 $a = 2, r = \frac{3}{8}$

3 $a = \frac{3}{2}, r = \frac{1}{2}$

4 $a = 3, r = \frac{1}{4}$

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Solution

Q. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 34, then [2000]

1 a=47, r=37

2 a=2, r=38

3 a=32, r=12

4 a=3, r=14