Sequences & Series
GP Sum Range — a²+b²
nta_pyq_2026_jan
Grade 11
Question:
In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a,b)$, then $a^2+b^2$ is equal to _____.
Step-by-Step Solution
Key Concept: Let terms be $3/r, 3, 3r$ (product $=27$, middle term $=3$). Sum $=3(1/r+1+r)$. For $r>0$: $r+1/r\geq2$, sum $\geq9$. For $r<0$: $r+1/r\leq-2$, sum $\leq-3$.
$a=-3$, $b=9$. $a^2+b^2=90$.
Correct Answer: 90