The value of∑(k=50)100k2is _________.a)338, 350b)297, 900c)297, 925d)290, 025Correct answer is option 'C'. Can you explain this answer?

Question

The value of&sum;(k=50)100k2is _________.a)338, 350b)297, 900c)297, 925d)290, 025Correct answer is option 'C'. Can you explain this answer?

Answer

Using the formula. ∑(k=1)ⁿ k² = (n(n + 1)(2n + 1)) / 6.

Answer

Understanding the Summation
To solve the problem, we need to evaluate the expression $\sum_{k=50}^{100} k^2$. This represents the sum of the squares of all integers from 50 to 100.
Formula for Sum of Squares
The formula for the sum of squares of the first $n$ integers is given by:
$$
\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}
$$
To find $\sum_{k=50}^{100} k^2$, we can calculate it using:
$$
\sum_{k=50}^{100} k^2 = \sum_{k=1}^{100} k^2 - \sum_{k=1}^{49} k^2
$$
Calculating Each Sum
1. Sum of Squares from 1 to 100:
- Here, $n = 100$:
$$
\sum_{k=1}^{100} k^2 = \frac{100(100 + 1)(2 \times 100 + 1)}{6} = \frac{100 \times 101 \times 201}{6} = 338350
$$
2. Sum of Squares from 1 to 49:
- Here, $n = 49$:
$$
\sum_{k=1}^{49} k^2 = \frac{49(49 + 1)(2 \times 49 + 1)}{6} = \frac{49 \times 50 \times 99}{6} = 40425
$$
Final Calculation
Now, subtract the two sums:
$$
\sum_{k=50}^{100} k^2 = 338350 - 40425 = 297925
$$
Thus, the value of $\sum_{k=50}^{100} k^2$ is 297,925, confirming that the correct answer is option C.

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