Evaluate the \(\sum_{n=3}^{12} \) -5n -1.

Solution:

Given, \(\sum_{n=3}^{12} \) -5n -1

k = 12 and n = 3,

Substituting, we get,

n = 3, -5(3) - 1 = -16

n = 4, -5(4) - 1 = -21

n = 5, -5(5)-1 = -26

n = 6, -5(6)-1 = -31

n = 7, -5(7)-1 = -36

n = 8, -5(8) - 1 = -41

n = 9, -5(9) - 1 = -46

n = 10, -5(10) - 1 = -51

n = 11, -5(11) - 1 = -56

n = 12, -5(12) - 1 = -61

There are 10 terms in the sequence. -16, -21, -26, -31, -36, -41, -46, -51 , -56, -61

This series is in arithmetic progression with a common difference of -5.

The first term is a = -16 and the last term l = -61

Summating all the values, we get

S_n = n(a+ l)/2

= 10 (-16 -61)/2

=5(-77)

= -385

Therefore, \(\sum_{n=3}^{12} \) -5n -1 = -385

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Evaluate the \(\sum_{n=3}^{12}\) -5n -1.

Summary:

On evaluating \(\sum_{n=3}^{12}\) -5n -1, we get -385.