If the number of ways of selecting 3 numbers out of 1, 2 , 3 , . . . . , 2 n + 1 such that they are in arithmetic progression is 441 , then the sum of the divisors of n is equal to

Step-by-step solution

If a , b , c are in A . P . , then a + c = 2 b i.e. the sum of two numbers is even ⇒ Both numbers are even or odd Odd numbers → 1,3 , 5 , . . . . . . , 2 n + 1 (total n + 1 ) Even numbers → 2,4 , 6 , . . . . . . , 2 n (total n ) ⇒ Required number of ways = n + 1 C 2 + n C 2 ⇒ n + 1 C 2 + n C 2 = 44 1 ⇒ n + 1 n 2 + n n - 1 2 = 44 1 ⇒ n 2 n + 1 + n - 1 = 44 1 ⇒ n 2 = 441 ⇒ n = 21 = 3 × 7 ⇒ Sum of divisors = 1 + 3 1 + 7 = 4 × 8 = 32