Question

The number of common terms in the two sequences: 15, 19, 23, 27, . . . . , 415 and 14, 19, 24, 29, . . . , 464 is

• A. 18
• B. 19
• C. 21
• D. 20

Answer 1

The Correct Option is D

Both sequences are in arithmetic progression.
The common difference \((d_1​)\) for the first sequence is 4.
The common difference \((d_2​)\) for the second sequence is 5.
The first common term is 19.
The common terms will also form an arithmetic progression with a common difference
\( LCM(d1​,d2​)=LCM(4,5)=20.\)

Let there be \(‘n’\) terms in this sequence; then, the last term would be less than or equal to 415.

i.e. \(a+(n−1)d≤415\) 
\(19+(n−1)×20≤415 \)
\((n−1)×20≤415−19 \)
\((n−1)×20≤396 \)
\((n−1)=[\frac{396}{20}​]\) where [ ] is the greatest integer 
\((n−1)=19, \)
so \(n=20\)