Progressions-7 (CAT-2008)

The number of common terms in the two sequences 17,21,25,….,417 and 16,21,26,….466 is

(1)78          (2) 19              (3)20                    (4)77               (5) 22

Solution follows here:

Solution:

Both the given series’ are in A.P.

Number of terms in the 1^st series: 417 = 17+(n₁-1)4 => n₁-1 = 400/4 = 100 => n₁=101

Number of terms in the 2^nd series: 466 = 16+(n₂-1)5 => n₂-1 = 450/5 = 90 => n₂=91

To find number of common terms:

Let m’th term of the first sequence = n’th term of the second sequence

Using formula for n’th term in an AP with initial term ‘a’ and common difference ‘d’,

t_n = a+(n-1)d,

17+(m-1)4 = 16+(n-1)5 => 4m-4+17 = 5n-5+16 => 4m+2 = 5n

=> we can observe that, m’th term of the first sequence is equal to the n’th term of the second sequence, for 

m=2, n=2;

m=7, n=6;

m=12, n=10;

… and so on…

m=97, n=78;

2^nd, 7^th, 12^th, …..97^th terms of the first series are respectively equal to the 2^nd, 6^th, 10^th, …..78^th terms of the second series. These are the common terms.

 Number of common terms: 97 = 2+(x-1)5 => x-1 = 95/5 = 19=> x=20

Answer (3)