IPMAT Indore 2025 · Quantitative Ability · Question 35

Step 1 — write general terms.

{a}_{k}=100+5\left( k-1 \right)=95+5k,{b}_{k}=100-5\left( k-1 \right)=105-5k

Step 2 — multiply them out.

{a}_{k}{b}_{k}=\left( 95+5k \right)\left( 105-5k \right)=9975-475k+525k-25{k}^{2}=9975+50k-25{k}^{2}

Step 3 — sum term by term over k=1 to 20, using

∑k=\frac{n\left( n+1 \right)}{2}=210, ∑{k}^{2}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}=\frac{20\cdot 21\cdot 41}{6}=2870

\sum_{k=1}^{20}{{a}_{k}}{b}_{k}=20\left( 9975 \right)+50\left( 210 \right)-25\left( 2870 \right)

Step 4 — evaluate.

=199500+10500-71750=138250

Answer: Option 2 — 138250