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  • Find the sum of the first n terms of the natural numbers, where the nth term is defined as <img alt="\mathrm{n^2+5 n-3.}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B

Find the sum of the first n terms of the natural numbers, where the nth term is defined as \mathrm{n^2+5 n-3.}

Option: 1

\mathrm{\left(n^3+5 n^2\right) / 2 .}


Option: 2

\mathrm{\left(n^3+5 n^2\right) / 2}


Option: 3

\mathrm{\left(n^3+4 n^2\right) / 2}


Option: 4

\mathrm{\left(2 n^3+7 n^2\right) / 2}


Answers (1)

best_answer

To find the sum of the first n terms of the natural numbers, where the nth term is defined as \mathrm{n^2+5 n-3, }we can use the formula for the sum of an arithmetic series.

The formula for the sum of an arithmetic series is given by:
 

\mathrm{ \text { Sum }=(n / 2)(\text { first term }+ \text { last term }) }

In this case, the first term is the value of the nth term when \mathrm{n=1}, which is \mathrm{\left(1^2+5(1)-3\right)=3.}

The last term is the value of the nth term when \mathrm{n=n}, which is \mathrm{\left(n^2+5 n-3\right).}

Substituting these values into the formula, we get:

\mathrm{ \operatorname{Sum}=(n / 2)\left(3+n^2+5 n-3\right) }

Simplifying the expression, we have:

\mathrm{ \text { Sum }=(n / 2)\left(n^2+5 n\right) }

Expanding further, we get:

\mathrm{ \text { Sum }=\left(n^3+5 n^2\right) / 2 }

Therefore, the sum of the first $\mathrm{n}$ terms of the natural numbers, where the nth term is defined as

\mathrm{n^2+5 n-3,\, \, is \, \, \left(n^3+5 n^2\right) / 2.}

Posted by

Irshad Anwar

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