Answer

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Hint- In order to solve the question, identify the type of series. Since it is an A.P. directly use the formula for sum of terms of arithmetic progression.

Complete step-by-step solution -

Given series: \[\left( {51 + 52 + 53 + ....... + 100} \right)\]

As by viewing the series it can be understood that the following series is an arithmetic progression series. Since the common difference between every two consecutive terms is the same and equal to 1.

For the given A.P.

First term of the series $ = a = 51$

Last term of the series $ = l = 100$

Number of terms in the series $ = n = 50$

As we know that sum of n terms of an A.P. with a as the first term and l as the last term is given by:

$S = \dfrac{n}{2}\left( {a + l} \right)$

So putting the values for the given series in the formula, we obtain the sum of the series.

\[

\Rightarrow {\text{Sum}} = \dfrac{n}{2}\left( {a + l} \right) \\

\Rightarrow {\text{Sum}} = \dfrac{{50}}{2}\left( {51 + 100} \right) \\

\Rightarrow {\text{Sum}} = \dfrac{{50 \times 151}}{2} \\

\Rightarrow {\text{Sum}} = 25 \times 151 \\

\Rightarrow {\text{Sum}} = 3775 \\

\]

Hence, the sum of the series is 3775.

So, option A is the correct option.

Note- In order to solve the problem related to sum of the terms in series. First and the basic step is to identify the type of series. The following formula of sum of n terms of A.P. is the simplest when the last term of the series is given else use the formula related to number of terms and common difference. This problem can also be solved by finding the sum of 100 terms of the series and subtracting the sum of the first 50 terms of the same series.

Complete step-by-step solution -

Given series: \[\left( {51 + 52 + 53 + ....... + 100} \right)\]

As by viewing the series it can be understood that the following series is an arithmetic progression series. Since the common difference between every two consecutive terms is the same and equal to 1.

For the given A.P.

First term of the series $ = a = 51$

Last term of the series $ = l = 100$

Number of terms in the series $ = n = 50$

As we know that sum of n terms of an A.P. with a as the first term and l as the last term is given by:

$S = \dfrac{n}{2}\left( {a + l} \right)$

So putting the values for the given series in the formula, we obtain the sum of the series.

\[

\Rightarrow {\text{Sum}} = \dfrac{n}{2}\left( {a + l} \right) \\

\Rightarrow {\text{Sum}} = \dfrac{{50}}{2}\left( {51 + 100} \right) \\

\Rightarrow {\text{Sum}} = \dfrac{{50 \times 151}}{2} \\

\Rightarrow {\text{Sum}} = 25 \times 151 \\

\Rightarrow {\text{Sum}} = 3775 \\

\]

Hence, the sum of the series is 3775.

So, option A is the correct option.

Note- In order to solve the problem related to sum of the terms in series. First and the basic step is to identify the type of series. The following formula of sum of n terms of A.P. is the simplest when the last term of the series is given else use the formula related to number of terms and common difference. This problem can also be solved by finding the sum of 100 terms of the series and subtracting the sum of the first 50 terms of the same series.

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