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Hint: Here, we will proceed by considering all the nth terms given in the options and then will put n=1,2,3,.. in order to find the reassemble of the obtained values of the terms with those of the given series.

Complete step-by-step answer:

The given sum of series is 3+7+13+21+…..

Let us consider all the nth terms given by obtains mentioned in the problem.

Consider the nth term as ${a_n} = 4n - 1$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_1} = \left( {4 \times 1} \right) - 1 = 4 - 1 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_2} = \left( {4 \times 2} \right) - 1 = 8 - 1 = 7$

Put n=3 for the third term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_3} = \left( {4 \times 3} \right) - 1 = 12 - 1 = 11$

But, since the third term of the given series is 13 which is not equal to 11 i.e., the third term of the series whose nth term is given by ${a_n} = 4n - 1$.

So, option A is incorrect.

Consider the nth term as ${a_n} = {n^n} + 2n$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^n} + 2n$, we get

${a_1} = {\left( 1 \right)^1} + \left( {2 \times 1} \right) = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^n} + 2n$, we get

${a_2} = {\left( 2 \right)^2} + \left( {2 \times 2} \right) = 4 + 4 = 8$

But, since the second term of the given series is 7 which is not equal to 8 i.e., the second term of the series whose nth term is given by ${a_n} = {n^n} + 2n$.

So, option B is also incorrect.

Consider the nth term as ${a_n} = {n^2} + n + 1$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_1} = {\left( 1 \right)^2} + 1 + 1 = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_2} = {\left( 2 \right)^2} + 2 + 1 = 4 + 3 = 7$

Put n=3 for the third term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_3} = {\left( 3 \right)^2} + 3 + 1 = 9 + 4 = 13$

Here, these values are the same as those of the given series. Hence, the nth term of the given series is given by ${a_n} = {n^2} + n + 1$.

So, option C is correct.

Consider the nth term as ${a_n} = {n^2} + 2$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^2} + 2$, we get

${a_1} = {\left( 1 \right)^2} + 2 = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^2} + 2$, we get

${a_2} = {\left( 2 \right)^2} + 2 = 4 + 2 = 6$

But, since the second term of the given series is 7 which is not equal to 6 i.e., the second term of the series whose nth term is given by ${a_n} = {n^2} + 2$.

So, option D is incorrect.

Note: In these types of problems, we eliminate the wrong options by putting different values of n starting from 1 in order to obtain first, second and so on up to that value of n which comes out to be different from those in the given series.

Complete step-by-step answer:

The given sum of series is 3+7+13+21+…..

Let us consider all the nth terms given by obtains mentioned in the problem.

Consider the nth term as ${a_n} = 4n - 1$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_1} = \left( {4 \times 1} \right) - 1 = 4 - 1 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_2} = \left( {4 \times 2} \right) - 1 = 8 - 1 = 7$

Put n=3 for the third term of the series whose nth term is given by ${a_n} = 4n - 1$, we get

${a_3} = \left( {4 \times 3} \right) - 1 = 12 - 1 = 11$

But, since the third term of the given series is 13 which is not equal to 11 i.e., the third term of the series whose nth term is given by ${a_n} = 4n - 1$.

So, option A is incorrect.

Consider the nth term as ${a_n} = {n^n} + 2n$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^n} + 2n$, we get

${a_1} = {\left( 1 \right)^1} + \left( {2 \times 1} \right) = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^n} + 2n$, we get

${a_2} = {\left( 2 \right)^2} + \left( {2 \times 2} \right) = 4 + 4 = 8$

But, since the second term of the given series is 7 which is not equal to 8 i.e., the second term of the series whose nth term is given by ${a_n} = {n^n} + 2n$.

So, option B is also incorrect.

Consider the nth term as ${a_n} = {n^2} + n + 1$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_1} = {\left( 1 \right)^2} + 1 + 1 = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_2} = {\left( 2 \right)^2} + 2 + 1 = 4 + 3 = 7$

Put n=3 for the third term of the series whose nth term is given by ${a_n} = {n^2} + n + 1$, we get

${a_3} = {\left( 3 \right)^2} + 3 + 1 = 9 + 4 = 13$

Here, these values are the same as those of the given series. Hence, the nth term of the given series is given by ${a_n} = {n^2} + n + 1$.

So, option C is correct.

Consider the nth term as ${a_n} = {n^2} + 2$.

Put n=1 for the first term of the series whose nth term is given by ${a_n} = {n^2} + 2$, we get

${a_1} = {\left( 1 \right)^2} + 2 = 1 + 2 = 3$

Put n=2 for the second term of the series whose nth term is given by ${a_n} = {n^2} + 2$, we get

${a_2} = {\left( 2 \right)^2} + 2 = 4 + 2 = 6$

But, since the second term of the given series is 7 which is not equal to 6 i.e., the second term of the series whose nth term is given by ${a_n} = {n^2} + 2$.

So, option D is incorrect.

Note: In these types of problems, we eliminate the wrong options by putting different values of n starting from 1 in order to obtain first, second and so on up to that value of n which comes out to be different from those in the given series.

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