Evaluate the summation of negative 5 n minus 1, from n, equals 3 to 12.

Solution:

We have to find the summation of negative 5 n minus 1, from n equals 3 to 12.

We can write it as

-5n - 1 where n = 3 to 12.

Substituting the values if

n = 3 = -16

n = 4 = -21

n = 5 = -26

n = 6 = -31

n = 7 = -36

n = 8 = -41

n = 9 = -46

n = 10 = -51

n = 11 = -56

n = 12 = -61

We get the 10 terms as   -16, -21, -26, -31, -36,-41,-46, -51, -56, -61

The first term is a₁= a = -16

The first term is a₁₂ = l = -61

This being in the arithmetic progression, with a common difference of -5, we can have the formula to find the sum of n terms in AP

Sum (S_n) = n(l+a)/2

S₁₀ = 10(-61-16)/2

= 5(-77)

= -385

Therefore, the summation is -385.

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Evaluate the summation of negative 5 n minus 1, from n equals 3 to 12.

Summary:

The summation of negative 5 n minus 1, from n equals 3 to 12 is -385.