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

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

Option: 1

\mathrm{\left(5 n^3-3 n^2+9 n\right) / 2}


Option: 2

\mathrm{\left(5 n^3-2 n^2+7 n\right) / 2}


Option: 3

\mathrm{\left(5 n^3+2 n^2+9 n\right) / 2}


Option: 4

\mathrm{\left(5 n^3-2 n^2+9 n\right) / 2}


Answers (1)

best_answer

To find the sum of the first n terms of the natural numbers, where each term is defined as \mathrm{5 n^2-2 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{ 5(1)^2-2(1)+3=6.}

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

Substituting these values into the formula, we get:

\mathrm{ \text { Sum }=(n / 2)\left(6+5 n^2-2 n+3\right) }

Simplifying the expression, we have:

\mathrm{ \text { Sum }=(n / 2)\left(5 n^2-2 n+9\right) }

Expanding further, we get:

\mathrm{ \text { Sum }=\left(5 n^3-2 n^2+9 n\right) / 2 }

Therefore, the value of the sum of the first $n$ terms of the natural numbers, where each term is defined as \mathrm{5 n^2-2 n+3}, is \mathrm{ \left(5 n^3-2 n^2+9 n\right) / 2.}

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SANGALDEEP SINGH

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