Question

Five real numbers a1, a2, a3, a4, and a5 are such that

√(a1 - 1 ) + 2√(a2 - 4) + 3√(a3 - 9) + 4√(a4 - 16) + 5√(a5 - 25) = (a1 + a2 + a3 + a4 + a5)/2.

Find a1 + a2 + a3 + a4 + a5.

Answer 1

$\sqrt{a_1-1}+2\sqrt{a_2-4}+3\sqrt{a_3-9}+4\sqrt{a_4-16}+5\sqrt{a_5-25}$ = $\frac{a_1+a_2+a_3+a_4+a_5}{2}$

or, $\sqrt{a_1-1}+\sqrt{4a_2-16}+\sqrt{9a_3-81}+\sqrt{16a_4-256}+\sqrt{25a_5-625}$ = $\frac{a_1}{2}+\frac{a_2}{2}+\frac{a_3}{2}+\frac{a_4}{4}+\frac{a_5}{2}$

or,

as five real numbers , $a_1,a_2,a_3,a_4$ and $a_5$

so, we can assume $\sqrt{a_1-1}=\frac{a_1}{2}$
or, $4a_1-4=a_1^2$

or, $a_1^2-4a_1+4=0\implies a_1=2$

similarly, $\sqrt{4a_2-16}=\frac{a_2}{2}$

or, $16a_2-64=a_2^2$

or, $a_2^2-16a_2+64=0\implies a_2=8$

$\sqrt{9a_3-81}=\frac{a_3}{2}$

or, $36a_3-324=a_3^2$

or, $a_3^2-36a_3+324=0\implies a_3=18$

similarly, $a_4=32$ and $a_5=50$

so, $a_1+a_2+a_3+a_4+a_5=2+8+18+32+50$
= 110.

Answer 2

a1 + a2 + a3 + a4 + a5 = 110