Find an equation for the n^th term of the arithmetic sequence. a₁₉ = -58, a₂₁ = -164

Solution:

It is given that

a₁₉ = -58 and a₂₁ = -164

We know that

a_n = a₁ + d(n - 1)

It can be written as

-58 = a₁ + d(19 - 1) 

-58 = a₁ + 18d ------- (1)

-164 = a₁ + d(21 - 1)

-164 = a₁ + d ------ (2)

Now multiply equation (1) by -1

58 = -a₁ - 18d

-164 = a₁ + 20d

By adding both equation ,we get

-106 = 2d

d = -53

Substituting the d value in either of the equations

-58 = a₁ + -53(18)

-58 = a₁ - 954

a₁ = -58 + 954

So we get,

a₁ = 896

The equation for n^th term is

a_n = 896 - 53(n - 1)

Therefore, the equation for the n^th term is a_n = 896 - 53(n - 1).

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Find an equation for the n^th term of the arithmetic sequence. a₁₉ = -58, a₂₁ = -164

Summary:

The equation for the n^th term of the arithmetic sequence when a₁₉ = -58, a₂₁ = -164 is a_n = 896 - 53(n - 1).