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  • Find the sum of the first 10 terms of the PRIME integers, where the nth term is defined as  <img alt="\mathrm{n^3+2 n \text {. }}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathr

Find the sum of the first 10 terms of the PRIME integers, where the nth term is defined as  \mathrm{n^3+2 n \text {. }}

Option: 1

3135

 


Option: 2

1002

 


Option: 3

2600

 


Option: 4

3212


Answers (1)

best_answer

To find the sum of the first 10 terms of the sequence generated by the expression \mathrm{n^3+2 n}, we need to substitute the values of n from 1 to 10 into the expression and sum them up.

The sum of the first 10 terms can be calculated as follows:

\mathrm{ \text { Sum }=1^3+2^3+3^3+\ldots+10^3+2(1+2+3+\ldots+10) }
The sum of the cubes of the first 10 natural numbers is given by the formula for the sum of cubes:

\mathrm{ 1^3+2^3+3^3+\ldots+n^3=(n(n+1) / 2)^2 }

Using this formula, we can calculate the sum of the cubes of the first 10 natural numbers:

Sum of cubes =(10(10+1) / 2)^2=(10(11) / 2)^2=(55)^2=3025

The sum of the first 10 natural numbers can be calculated using the formula for the sum of an arithmetic series:

1+2+3+\ldots+n=(n(n+1)) / 2
Using this formula, we can calculate the sum of the first 10 natural numbers:

Sum of 1 to 10=(10(10+1)) / 2=(10(11)) / 2=55

Multiplying the sum of the first 10 natural numbers by 2 , we get:

2(1+2+3+\ldots+10)=2(55)=110

Adding the sum of the cubes of the first 10 natural numbers to 2 times the sum of the first 10 natural numbers, we get the final sum:

\text { Sum }=3025+110=3135

Therefore, the sum of the first 10 terms of the sequence generated by \mathrm{n^3+2 n\, \, is \, \, 3135 .}

Posted by

Pankaj Sanodiya

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