If \ a _ i \ _ i = 1 ^ n , where n is an even integer, is an...

Given, $\mathop{∑}_{i = 1}^{n}{a}_{i} = 192$

${a}_{1} + {a}_{2} ⋯ {a}_{n} = 192$

$⇒ \dfrac{n}{2}\left({a}_{1} + {a}_{n}\right) = 192$

${a}_{1} + {a}_{n} = \dfrac{384}{n} ⋯ \left(1\right)$

Also given $\mathop{∑}_{i = 1}^{\dfrac{n}{2}}{a}_{2i} = 120$

$⇒ \mathop{\mathop{{a}_{2} + {a}_{4} + {a}_{6} ⋯ {a}_{n}}_{⏟}}_{\dfrac{n}{2}terms} = 120$

$⇒ \dfrac{n}{2} \times \dfrac{1}{2}\left[{a}_{2} + {a}_{n}\right] = 120$

${a}_{2} + {a}_{n} = \dfrac{480}{n}$

${a}_{1} + 1 + {a}_{n} = \dfrac{480}{n} ⋯ \left(2\right)$

Now equation \left(2\right) -$equation \left(1\right)$

$\dfrac{480}{n} - \dfrac{384}{n} = \left({a}_{1} + {a}_{n} + 1\right) - \left(a, + {a}_{n}\right)</p><p>\dfrac{1}{n}\left(480 - 384\right)$ = 1$</p><p>$480 - 384 = n$</p><p>$n = 96$</p>$