1. Series Problem

An and Bn are two series given by:
An = $\displaystyle 1^2$+ $\displaystyle 5^2$ + $\displaystyle 9^2$+ $\displaystyle 13^2$ + …… + (4n–3$\displaystyle )^2$
Bn = $\displaystyle 3^2$ + $\displaystyle 7^2$ + $\displaystyle 11^2$ + $\displaystyle 15^2$ + ….
For n = 1, 2, 3,….
i. Find the nth term of Bn.
Tn = $\displaystyle (4n - 1)^2$

ii. If Sn = An – Bn, prove that S2n = $\displaystyle -8n^2$

i have so far for this only
Sn = $\displaystyle 1^2$+ $\displaystyle 5^2$ + $\displaystyle 9^2$+ $\displaystyle 13^2$ + …… + (4n–3$\displaystyle )^2$
- ($\displaystyle 3^2$ + $\displaystyle 7^2$ + $\displaystyle 11^2$ + $\displaystyle 15^2$ + $\displaystyle (4n - 1)^2$)

and have absolutely no idea how to proceed from here

iii. Hence, evaluate $\displaystyle 101^2$ – $\displaystyle 103^2$ + $\displaystyle 105^2$ – $\displaystyle 107^2$ + …… + $\displaystyle 1993^2$ - $\displaystyle 1995^2$

any help greatly appreciated
thanks

2. Originally Posted by jacs
An and Bn are two series given by:
An = $\displaystyle 1^2$+ $\displaystyle 5^2$ + $\displaystyle 9^2$+ $\displaystyle 13^2$ + …… + (4n–3$\displaystyle )^2$
Bn = $\displaystyle 3^2$ + $\displaystyle 7^2$ + $\displaystyle 11^2$ + $\displaystyle 15^2$ + ….
For n = 1, 2, 3,….
i. Find the nth term of Bn.
Tn = $\displaystyle (4n - 1)^2$

ii. If Sn = An – Bn, prove that S2n = $\displaystyle -8n^2$

$\displaystyle A_n=\sum_{r=1}^n (4r-3)^2=\sum_{r=1}^n \left[16r^2-24r+9\right]$

You should know the formula for the sum of the first $\displaystyle n$ integers and their squares.

Same for $\displaystyle B_n$

CB

3. Thanks CB
went ahead with the sigma and tried this, although i must be doing something wrong because i cannot get the required result. (pls see attached image)

4. Originally Posted by jacs
Thanks CB
went ahead with the sigma and tried this, although i must be doing something wrong because i cannot get the required result. (pls see attached image)
I agree, the given answer is $\displaystyle S_n$ not $\displaystyle S_{2n}$ . To show this compute $\displaystyle S_1=A_1-B_1$ and $\displaystyle S_2=A_2-B_2$.
CB