CAT 2008 — QA Question 4
The number of common terms in the two sequences 17, 21, 25, … , 417 and 16, 21, 26, … , 466 is
Answer & solution
Correct answer: 20
- A
78
- B
19
20
- D
77
- E
22
The first sequence can be written as 17, 17 + 4, 17 + 8, … , 417 and second sequence can be written as 16, 16 + 5, 16 + 10, … , 466
The common difference for the first sequence is 4 and that for the second sequence is 5 and both the sequences have 21 as the first common term.
∴ Common terms are 21, 21 + L, 21 + 2L, ...
[Here, L = LCM of 4 and 5 = 20]
∴ Common terms are 21, 21 + 20, 21 + 40, ...
The common terms have a common difference of 20 and first term as 21.
∴ Let Tn be the last common term, Tn = 21 + (n - 1) × 20
Now, Tn should be less than or equal to 417
⇒ Tn = 21 + (n - 1) × 20 ≤ 417
⇒ 20n - 20 ≤ 396
⇒ n ≤ 416/20 = 20.8
Hence, the highest possible value of n is 20.
∴ The total number of terms which are common to both the sequences = 20
Hence, option (c).