Question

The number of common terms to the two sequences $17,21,25,\cdots ,417$ and $16,21,26,\cdots ,466$, is

• A. $19$
• B. $20$
• C. $21$
• D. $91$

Answer 1

Answer: (B)

$20$

Consider the terms $17,21,25\dots 417$ where $a=17$ and $d=21-17=4$

Total number of terms in the sequence

$n=\frac{l-a}{d}+1=\frac{417-17}{4}+1=100+1=101$

Consider the terms $16,21,26\dots 466$ where $a=16$ and $d=21-16=5$

Total number of terms in the sequence

$n=\frac{l-a}{d}+1=\frac{466-16}{5}+1=90+1=91$

$n^{th}$ term of the first sequence $17+(n-1)4=4n+13$

$m^{th}$ term of the second sequence $16+(m-1)5=5m+11$

Common terms$=4n+13=5m+11$

$\Rightarrow 5m-4n=2$

Possible integral values of n that satisfy $5m=2+4n$ are $\{2,7,12\dots 97\}$ where $a=2,d=7-2=5,l=97$

The total number of terms common in both the sequences is $\frac{97-2}{5}+1=\frac{95}{5}+1=19+1=20$ terms.