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  • Calculate the sum of the first $n$ terms of the natural numbers, where the nth term is given by <img alt="\mathrm{3 n+2.}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B3%20n&amp

Calculate the sum of the first $n$ terms of the natural numbers, where the nth term is given by \mathrm{3 n+2.}

Option: 1

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


Option: 2

\mathrm{(n / 2)(3 n+2)}


Option: 3

\mathrm{(n / 2)(5 n+7)}


Option: 4

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


Answers (1)

best_answer

To calculate the sum of the first n terms of the natural numbers, where the nth term is given by \mathrm{ 3 n+2}, we need to substitute the values of n from 1 to n into the expression and sum them up.

The sum of the first $\mathrm{n}$ terms can be calculated as follows:

\text { Sum }=(3(1)+2)+(3(2)+2)+\ldots+(3(n)+2)

Simplifying the expression for each term, we have:

\text { Sum }=(3+2)+(6+2)+\ldots+(3 n+2)

Simplifying further, we get:

\text { Sum }=5+8+\ldots+(3 n+2)

To find the sum of an arithmetic series, we can use the formula:

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

In this case, the first term is 5 , the last term is \mathrm{(3 n+2)}, and the number of terms is n.
Plugging these values into the formula, we get:

\text { Sum }=(n / 2)(5+3 n+2)=(n / 2)(3 n+7)

Therefore, the sum of the first n terms of the natural numbers, where the nth term is given by \mathrm{ 3 n+2,} is \mathrm{ (n / 2)(3 n+7)}

Posted by

Ritika Kankaria

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