An algebra problem by Akash singh

$\begin{aligned}\begin{aligned}\color{#3D99F6}A=&\ \color{#D61F06}{3^2}+\color{#20A900}{4^2}+\color{#D61F06}{7^2}+\color{#20A900}{8^2}+\color{#D61F06}{11^2}+\color{#20A900}{12^2}+\cdots+\color{#D61F06}{99^2}+\color{#20A900}{100^2}\\=&\ \color{#D61F06}{(3^2+7^2+11^2+\cdots+99^2)}+\color{#20A900}{(4^2+8^2+12^2+\cdots+100^2)}\\=&\ \color{#D61F06}{\displaystyle\sum_{n=1}^{25}(4n-1)^2}+\color{#20A900}{\displaystyle\sum_{n=1}^{25}(4n)^2}\\=&\ \color{#D61F06}{\displaystyle\sum_{n=1}^{25}(16n^2-8n+1)}+\color{#20A900}{\displaystyle\sum_{n=1}^{25}16n^2}\\=&\ \displaystyle\sum_{n=1}^{25}(16n^2-8n+1+16n^2)\\=&\ \color{#3D99F6}{\displaystyle\sum_{n=1}^{25}(32n^2-8n+1)}\end{aligned}\end{aligned}$

$\begin{aligned}\begin{aligned}\color{#302B94}B=&\ \color{#624F41}{1^2}+\color{magenta}{2^2}+\color{#624F41}{5^2}+\color{magenta}{6^2}+\color{#624F41}{9^2}+\color{magenta}{10^2}+\cdots+\color{#624F41}{97^2}+\color{magenta}{98^2}\\=&\ \color{#624F41}{(1^2+5^2+9^2+\cdots+97^2)}+\color{magenta}{(2^2+6^2+10^2+\cdots+98^2)}\\=&\ \color{#624F41}{\displaystyle\sum_{n=1}^{25}(4n-3)^2}+\color{magenta}{\displaystyle\sum_{n=1}^{25}(4n-2)^2}\\=&\ \color{#624F41}{\displaystyle\sum_{n=1}^{25}(16n^2-24n+9)}+\color{magenta}{\displaystyle\sum_{n=1}^{25}(16n^2-16n+4)}\\=&\ \displaystyle\sum_{n=1}^{25}(16n^2-24n+9+16n^2-16n+4)\\=&\ \color{#302B94}{\displaystyle\sum_{n=1}^{25}(32n^2-40n+13)}\end{aligned}\end{aligned}$

Thus,

$\begin{aligned}\begin{aligned}\color{#3D99F6}A-\color{#302B94}B=&\ \color{#3D99F6}{\displaystyle\sum_{n=1}^{25}(32n^2-8n+1)}-\color{#302B94}{\displaystyle\sum_{n=1}^{25}(32n^2-40n+13)}\\=&\ \displaystyle\sum_{n=1}^{25}(32n^2-8n+1-32n^2+40n-13)\\=&\ \displaystyle\sum_{n=1}^{25}(32n-12)\\=&\ 32\displaystyle\sum_{n=1}^{25}n-\displaystyle\sum_{n=1}^{25}12\\=&\ 32\left(\frac{25(26)}2\right)-25(12)\\=&\ 16(650)-300\\=&\ 10400-300=\boxed{10100}\end{aligned}\end{aligned}$

here n sq=n*n

A-B= 3sq - 1sq + 4sq - 2sq + 7sq - 5sq + 8sq - 6sq +.......................99sq - 97sq + 100sq - 98sq

using asq-bsq =(a+b)(a-b)

A-B=(3-1)(3+1)+(4-2)(4+2)+(7-5)(7+5)+(8-6)(8+6)+...............+(99-97)(99+97)+(100-98)(100+98)

=2[1+2+3+4+5+6+ . . . . . . . . . . . . . . . . . . +97+98+99+100]=2[(100)(101)]/2 =10100