A sequence of terms U1, U2, U3, ... is defined by Un = 5n + 20
If the sum of the first N terms of the sequence defined above equals the sum of the first kN natural numbers, show that:
N = 45 - k / k^2 - 5
Sum of first N natural numbers = N(N+1)/2
Sum of sequence = N(50+(N-1)5)/2 (knowing that a=25, d=5)
The Attempt at a Solution
I put the substituted kN into N in the equation N(N+1)/2 to give me kN(kN+1)/2.
I then made kN(kN+1)/2 = N(50+(N-1)5)/2 and solved from there which eventually got me to:
k^2N^2 + kN = 50N + 5N^2 -5
(I then divided both sides by N as N is always positive)
N = 45 +5N -k / k^2
...the +5N makes it wrong. I dont know where I've gone wrong, any help would be appreciated.