Series Question (1 Viewer)

[Ostentatious]

Find S_10 of: 3 + 7 + 13 + … + [2^n + (2n – 1)] + …

 

[Trebla]

S_n = 3 + 7 + 13 + .......... + [2ⁿ + (2n - 1)]
= (2 + 1) + (4 + 3) + (8 + 5) + (16 + 7) + ......... + [2ⁿ + (2n - 1)]
= (2 + 4 + 8 + 16 + ......... + 2ⁿ) + (1 + 3 + 5 + 7 + .......... + (2n - 1))
The first part is a GP, and the second part is an AP.
= 2(2ⁿ - 1) / (2 - 1) + n(1 + 2n - 1)/2
= 2^{n + 1} - 2 + n²
When n = 10
S₁₀ = 2¹¹ - 2 + 100
= 2146

 

[Ostentatious]

Thank you so much! I knew this was a mixed AP GP question, I just didn't realise I had to go about it this way. You're working was nice and clear