
Find the sum of the series \[{11^2} + {12^2} + {13^2} + ... + {20^2}\] .
A.2481
B.2483
C.2485
D.2487
Answer
162k+ views
Hint: Observe that in the given series there are 10 terms but our formula starts from 1 so first obtain the sum of square of first 20 terms then subtract the sum of first ten term to obtain the result.
Formula Used: The formula of sum of n square terms \[{1^2} + {2^2} + {3^2} + ... + {n^2}\] is \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] .
Complete step by step solution: Observe that in the given series there are 10 terms but our formula starts from 1 so first obtain the sum of square of first 20 terms then subtract the sum of first ten term to obtain the result.
Substitute 20 for n in the formula \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] and calculate to obtain the required result.
\[\dfrac{{20(20 + 1)(2.20 + 1)}}{6}\]
\[ = \dfrac{{20 \times 21 \times 41}}{6}\]
=2870
Substitute 10 for n in the formula \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] and calculate to obtain the required result.
\[\dfrac{{10(10 + 1)(2.10 + 1)}}{6}\]
\[ = \dfrac{{10 \times 11 \times 21}}{6}\]
=385
Subtract 385 from 2870 to obtain the required result.
\[2870 - 385 = 2485\]
Option ‘C’ is correct
Note: Sometime students directly substitute 10 for n and calculate the result but observe that our formula start from 1 so we need to calculate the sum of square of first 20 terms first then have to subtract the sum of square of first 10 terms to obtain the required result.
Formula Used: The formula of sum of n square terms \[{1^2} + {2^2} + {3^2} + ... + {n^2}\] is \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] .
Complete step by step solution: Observe that in the given series there are 10 terms but our formula starts from 1 so first obtain the sum of square of first 20 terms then subtract the sum of first ten term to obtain the result.
Substitute 20 for n in the formula \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] and calculate to obtain the required result.
\[\dfrac{{20(20 + 1)(2.20 + 1)}}{6}\]
\[ = \dfrac{{20 \times 21 \times 41}}{6}\]
=2870
Substitute 10 for n in the formula \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] and calculate to obtain the required result.
\[\dfrac{{10(10 + 1)(2.10 + 1)}}{6}\]
\[ = \dfrac{{10 \times 11 \times 21}}{6}\]
=385
Subtract 385 from 2870 to obtain the required result.
\[2870 - 385 = 2485\]
Option ‘C’ is correct
Note: Sometime students directly substitute 10 for n and calculate the result but observe that our formula start from 1 so we need to calculate the sum of square of first 20 terms first then have to subtract the sum of square of first 10 terms to obtain the required result.
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