The sum of the series $(2)^2+2(4)^{2}+3(6)^{2}+....$ upto $10$ terms is
- $12100$
- $11300$
- $11200$
- $12300$
Reveal answer
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A
Correct answer
Explanation
The general term of the series is T_n = n * (2n)^2 = 4n^3. The sum of the first 10 terms is therefore 4 times the sum of the cubes of the first 10 natural numbers. Using the formula for the sum of cubes, we get 4 * (10 * 11 / 2)^2 = 4 * 3025 = 12100.
AI explanation
The nth term of the series follows the pattern n multiplied by (2n) squared, which simplifies to 4n^3. To find the sum of the first 10 terms, we calculate 4 times the sum of the cubes of the first 10 natural numbers using the formula for the sum of cubes. The sum of cubes is [10(11)/2]^2 = 55^2 = 3025, and multiplying this by 4 yields 12100. The result is 12100.