The sum of the first 24 terms of the series \(\rm \sqrt{2}+ \sqrt{8}+\sqrt{18}+\sqrt{32}+\ ...\) is:

Question

Question

\(100\sqrt{2}\)
\(300\sqrt{2}\)
\(200\sqrt{2}\)
\(500\sqrt{2}\)
None of the above / More than one of the above

Answer 1

The correct answer is \(300\sqrt{2}\)

Concept:

Sum of consecutive numbers from 1 to n: \(\rm1 + 2 + 3 +\ ...\ + n = \dfrac{n(n+1)}{2}\).

Calculation:

The sum of the first 24 terms of the given series can be written as:

\(\rm \sqrt{2}+ \sqrt{8}+\sqrt{18}+\sqrt{32}+\ ...\)

\(\rm =\sqrt{2}+ 2\sqrt{2}+3\sqrt{2}+4\sqrt{2}+\ ...\ +24\sqrt2\)

\(\rm =\sqrt2(1+2+\ ...\ +24)\)

\(\rm =\sqrt2\times \dfrac{24\times25}{2}=300\sqrt2\).