
2. Let $a_{1}, a_{2}, \ldots, a_{2 n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let
(i) $a_{1}^{2}+a_{3}^{2}+\cdots+a_{2 n-1}^{2}=x$,
(ii) $a_{2}^{2}+a_{4}^{2}+\cdots+a_{2 n}^{2}=y$, and
(iii) $a_{n}+a_{n+1}=z$.

Express $d$ in terms of $x, y, z, n$.


Solution: Observe that

$$
y-x=\left(a_{2}^{2}-a_{1}^{2}\right)+\left(a_{4}^{2}-a_{3}^{2}\right)+\cdots+\left(a_{2 n}^{2}-a_{2 n-1}^{2}\right)
$$

The general difference is

$$
a_{2 k}^{2}-a_{2 k-1}^{2}=\left(a_{2 k}+a_{2 k-1}\right) d=\left(2 a_{1}+((2 k-1)+(2 k-2)) d\right) d
$$

Therefore

$$
y-x=\left(2 n a_{1}+(1+2+3+\cdots(2 n-1)) d\right) d=n d\left(2 a_{1}+(2 n-1) d\right)
$$

We also observe that

$$
z=a_{n}+a_{n+1}=2 a_{1}+(2 n-1) d
$$

It follows that $y-x=n d z$. Hence $d=(y-x) / n z$.
