Evaluate the summation of 2 n plus 5, from n equals 1 to 12.

Solution:

Given, the expression is 2 n plus 5.

We have to find the summation of 2 n plus 5, from n equals 1 to 12.

Summation of 2 n plus 5 can be written as \(\sum_{n=1}^{12}2n+5\)

= 2(1) + 5 + 2(2) + 5 + 2(3) + 5………………….+ 2(12) + 5

= 2 + 5 + 4 + 5 + 6 + 5………………….+ 24 + 5

= [2 + 4 + 6 + ……… + 24] + 12(5)

= [2 + 4 + 6 + ……… + 24] + 60

[2+ 4 + 6 + ……… + 24] represents arithmetic progression.

The sum of an arithmetic progression is given by

S = n/2 [2a + (n - 1)d]

Where, a is the first term

d is the common difference

Here, a = 2, n = 12

Common difference, d = 4 -2 = 6 - 4 = 2

S = 12/2 [2(2) + (12 - 1)(2)]

= 6[4 + 11(2)]

= 6[4 + 22]

= 6(26)

[2 + 4 + 6 + ……… + 24] = 156

Now, [2 + 4 + 6 + ……… + 24] + 60 = 156 + 60

= 216

Therefore, the required value is 216.

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Evaluate the summation of 2 n plus 5, from n equals 1 to 12.

Summary:

The summation of 2 n plus 5, from n equals 1 to 12 is 216.