QJEEM24Ma682a: JEE Main 2024 Mathematics - Class 11 JEEM24Ma682a Solution: Let be in an arithmetic progression of positive terms. Let . If …

Let $a_n=a+(n-1) d \forall n \in N$

$\begin{aligned}A_k & =\left(a_1^2-a_2^2\right)+\left(a_3^2-a_4^2\right)+\ldots a_{2 k-1}^2-a_{2 k}^2 \\& =(-d)\left(a_1+a_2+\ldots+a_{2 k}\right) \\& A_k=(-d k)(2 a+(2 k-1) d) \\\Rightarrow & A_3=(-3 d)(2 a+5 d)=-153 \\\Rightarrow & d(2 a+5 d)=51 \quad \text{... (i)}\\& A_5=(-5 d)(2 a+9 d)=-435\end{aligned}$

$\begin{aligned}\Rightarrow & d(2 a+9 d)=87 \\\Rightarrow & 4 d^2=36 \Rightarrow d= \pm 3(d=3 \text { positive terms }) \\\Rightarrow & 3(2 a+27)=87 \\\Rightarrow & 2 a=29-27 \\\Rightarrow & a=1 \\& a_{17}-A_7=(a+16 d)-(-7 d)(2+13 d) \\& =49+7 \times 3(2+39) \\& =49+21 \times 41=910\end{aligned}$