Sequences and Series · Question 50

For positive integers n, if 4an = (n2 + 5n + 6 ) and Sn = k=1n ( 1ak ), then the value of 507S2025 is :

1. 540
2. 675
3. 1350
4. 135

Answer

(2) 675

Solution

an = (n + 2)(n + 3)4

1ak = 4(k + 2)(k + 3) = 4 ( 1k + 2 - 1k + 3 )

Sn = 4 ( 13 - 1n + 3 ) (telescoping)

S2025 = 4 ( 13 - 12028 ) = 4 676 - 12028 = 4 6752028

507S2025 = 507 4 6752028

= 675 (as 2028 = 4 507)

Answer: (2)