Question

The number of terms common between the two series $2+5+8+$... up to 50 terms and the series $3+5+7+9+$... up to 60 terms are

• A. $24$
• B. $26$
• C. $25$
• D. $20$

Answer 1

The correct option is D $20$

Consider the first AP $:2,5,8,11,14,17,20,\mathrm{23....}$

Thus, $a=2$ and $d=3$

Hence 50th term of the AP

$a_{50}=a+(n-1)d$

$=2+(50-1)3$

$=2+147$

$=149$

Consider the second AP $:3,5,7,9,11,13,17,19,21,\mathrm{23....}$

Thus, $a=3$ and $d=2$

Hence 60th term of the AP

$a_{60}=a+(n-1)d$

$=3+(60-1)2$

$=3+118$

$=121$

Consider the common terms of the above two AP's

$5,11,17,23,\mathrm{29...}$

Thus, $a=5$ and $d=6$

Since 121 is the last term of the second AP, the last term of the common AP will be less than or equal to 121.

Considering the last term of common AP as 121, we have

$a_n=a+(n-1)d$

$\Rightarrow 121=5+(n-1)6$

$\Rightarrow 121-5=(n-1)d$

$\Rightarrow 116=(n-1)6$

$\Rightarrow n-1=\frac{116}{6}$

$\Rightarrow n=19.33+1$

$\Rightarrow n=20.33$

As, 'n' cannot be fraction, taking the nearest whole number, we have $n=20$

Thusm there are 20 common terms of the two AP's.