Algebra

Progression & Series

Medium

The sum of 30 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + ...$ is

$\frac{930}{\sqrt{2}}$

$465$

$\frac{465}{\sqrt{2}}$

None of these

✅ Correct Option: 1

The series is: $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + ...$

Simplify each term:

$\sqrt{2} = \sqrt{2 \cdot 1} = \sqrt{2}$

$\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

$\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$

Each term is $n\sqrt{2}$ where $n = 1, 2, 3, 4, ...$

The series becomes: $\sqrt{2} + 2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} + ...$

Factor out $\sqrt{2}$ :

$\sqrt{2}(1 + 2 + 3 + 4 + ... + 30)$

Using the formula for the sum of first $n$ natural numbers:

$\dfrac{n(n+1)}{2}$

For $n = 30$ :

$1 + 2 + 3 + ... + 30 = \dfrac{30 \times 31}{2}$

$= \dfrac{930}{2}$

$= 465$

Sum of 30 terms:

$\sqrt{2} \times 465 = 465\sqrt{2}$

This can also be written as:

$\dfrac{930}{\sqrt{2}} = \dfrac{930\sqrt{2}}{2} = 465\sqrt{2}$

Therefore, the sum of 30 terms is $465\sqrt{2}$ or equivalently $\dfrac{930}{\sqrt{2}}$ .

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The sum of 30 terms of the series 2+8+18+32+...\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + ...2​+8​+18​+32​+... is

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The sum of 30 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + ...$ is