Arithmetic Sequence

Question

The following problem refers to an arithmetic sequence. If the fourth term is  19 and the eleventh term is 54, find the term a1, the common difference d, and then find a23 and S23

Answer 1

In an arithmetic sequence, the nth term, a_n, is given by the formula       a_n = a₁ + (n-1)d, where d is the common difference.

 

So,  a₁ +  3d = 19

and a₁ + 10d = 54

 

Subtract the equations to get -7d = -35

 

                                            d = 5

 

                                            a₁ + 3(5) = 19

 

                                            a₁ = 4

 

We have a_n  = 4 + 5(n-1)

 

So, a₂₃ = 4 + 5(22) = 4 + 110 = 114

 

S_n = sum of first n terms = (n/2)(a₁ + a_n)

 

So, S₂₃ = (23/2)(4+114) = 1357