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

Two arithmetic progressions (APs) are given. The first AP has a common difference \( d_1 = 4 \), and the second AP has a common difference \( d_2 = 5 \). The first common term in both APs is 19, and we want to find how many such common terms are there that are less than or equal to 415.

Step 1: Common Difference of Common Terms

Since the common terms must satisfy both APs, their common difference will be the least common multiple (LCM) of the individual differences: \[ \text{Common difference} = \text{LCM}(d_1, d_2) = \text{LCM}(4, 5) = 20 \]

Step 2: General Term of Common Sequence

The common sequence is an AP with:

• First term: \( a = 19 \)
• Common difference: \( d = 20 \)

Let the number of terms be \( n \). Then the \( n \)-th term is: \[ a + (n - 1)d \leq 415 \Rightarrow 19 + (n - 1) \cdot 20 \leq 415 \Rightarrow (n - 1) \cdot 20 \leq 396 \Rightarrow n - 1 \leq \left\lfloor \frac{396}{20} \right\rfloor = 19 \Rightarrow n = 20 \]

 

Final Answer: \( \boxed{20} \) common terms