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  • Calculate the sum of the first n terms of the natural numbers, where the nth term is expressed as <img alt="\mathrm{n^3-n^2+2 n-1.}" src="https://entrancecorner.oncodecogs.com/gif.latex?%

Calculate the sum of the first n terms of the natural numbers, where the nth term is expressed as \mathrm{n^3-n^2+2 n-1.}

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

\mathrm{\left(n^4-2 n^3+2 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 expressed as \mathrm{ n^3-n^2+2 n-1}, 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{ 1^3-1^2+2(1)-1=1.}

The last term is the value of the $n$th term when\mathrm{ \mathrm{n}=\mathrm{n}}, which is \mathrm{ n^3-n^2+2 n-1.}

Substituting these values into the formula, we get:

\mathrm{ \operatorname{Sum}=(n / 2)\left(1+n^3-n^2+2 n-1\right) }

Simplifying the expression, we have:

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

Expanding further, we get:

\mathrm{ \text { Sum }=\left(n^4-n^3+2 n^2\right) / 2 }
Therefore, the sum of the first n terms of the natural numbers, where each term is expressed as \mathrm{n^3-n^2+2 n-1, } is

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

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vishal kumar

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