Question

The first term of an infinite geometric series is $21$. The second term and the sum of the series are both positive integers. The possible value(s) of the second term can be

• A. $12$
• B. $14$
• C. $18$
• D. $20$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $12$
B $14$
C $18$
D $20$

Let first term$=21$ and common ratio$=r$

Then the terms of G.P are $21,21r,21r^2,...$ where $21r\in I^{+}$

$S=\frac{21}{1-r}$ where $1-r>0$ or $r<1$

$S=\frac{21\times 21}{21(1-r)}=\frac{21\times 21}{21-21r}$ where $21-21r=$integer.

$21-21r=1,3,7,9,21,\mathrm{63...}$

$1-r=1,3,7,21,..$

$r=0,-2,-6,-20,...$

At $21-21r=1\Rightarrow r=\frac{20}{21}$

$21r=21\times \frac{20}{21}=20$

At $21-21r=3\Rightarrow r=\frac{18}{21}=\frac{6}{7}$

$21r=21\times \frac{6}{7}=18$

At $21-21r=7\Rightarrow r=\frac{14}{21}=\frac{2}{3}$

$21r=21\times \frac{2}{3}=14$

At $21-21r=9\Rightarrow r=\frac{12}{21}=\frac{4}{7}$

$21r=21\times \frac{4}{7}=12$