The value of $5^2 + 6^2 + .... + 10^2 + 20^2$ is

### Concept & Formula To find the sum of squares of a continuous segment of natural numbers, we use the standard formula for the sum of the squares of the first $n$ natural numbers, and subtract the missing initial terms. Note that the final term $20^2$ acts as a disconnected standalone addition to the primary series. $$ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} $$ ### Step-by-Step Solution * **Given Series:** $(5^2 + 6^2 + .... + 10^2) + 20^2$ * **Calculation / Deduction:** First, find the sum of the continuous series from $5^2$ to $10^2$. This is equivalent to finding the sum from $1^2$ to $10^2$, and subtracting the sum from $1^2$ to $4^2$. Sum from $1$ to $10$ ($n=10$): $$ \frac{10(11)(21)}{6} = \frac{2310}{6} = 385 $$ Sum from $1$ to $4$ ($n=4$): $$ \frac{4(5)(9)}{6} = \frac{180}{6} = 30 $$ Sum from $5^2$ to $10^2$: $$ 385 - 30 = 355 $$ Now, add the standalone $20^2$ term at the end of the expression: $$ 20^2 = 400 $$ $$ Total = 355 + 400 = 755 $$ ### Exam Strategy & Shortcut **Unit Digit Check:** If you are short on time, calculate the unit digits of the squares mentally. $5^2 \rightarrow 5$ $6^2 \rightarrow 6$ $7^2 \rightarrow 9$ $8^2 \rightarrow 4$ $9^2 \rightarrow 1$ $10^2 \rightarrow 0$ $20^2 \rightarrow 0$ Summing these unit digits: $5 + 6 + 9 + 4 + 1 + 0 + 0 = 25$. The final unit digit must be $5$. Looking at the options, both $755$ and $765$ end in $5$, narrowing it down to a 50/50 guess, but calculating the actual arithmetic is safe and fast enough here. ### Common Pitfall Students often skim the question and mistakenly assume it is a continuous series from $5^2$ all the way to $20^2$. Paying close attention to the punctuation and exact numbers given is critical; the series explicitly stops at $10^2$ and then appends a singular $20^2$. ### Final Answer **Therefore, the correct answer is 755.**