IPMAT Indore 2025 · Quantitative Ability · Question 18

Step 1 — recognise the polynomial. Multiply by \left( x-1 \right):

\left( x-1 \right)\left({x}^{8}+{x}^{7}+\cdots +x+1 \right)={x}^{9}-1

So the roots of {x}^{8}+\cdots +1=0 are exactly the solutions of {x}^{9}=1 except x=1. These are the eight non-real ninth roots of unity.

Step 2 — use {a}_{i}^{9}=1. For every root, {a}_{i}^{9}=1. Now look at the exponent 2025:

2025=9\times 225

so 2025 is an exact multiple of 9. Therefore

{a}_{i}^{2025}={\left({a}_{i}^{9}\right)}^{225}={1}^{225}=1

Step 3 — add up all eight.

{a}_{1}^{2025}+\cdots +{a}_{8}^{2025}=\underset{8\text{times}}{\underset{⏟}{1+1+\cdots +1}}=8

Answer: Option 3 — 8