Find the general expression for the nth term of the arithmetic sequence

Question

Find the general expression for the nth term of the arithmetic sequence if a₈ = 45 and a₁₂ =69

Answer 1

Find the general expression for the nth term of the arithmetic sequence

Find the general expression for the nth term of the arithmetic sequence if a₈ = 45 and a₁₂ =69

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An arithmetic sequence has a common difference, d, between adjacent terms.

The problem gives: a₁₂ =69 so, 69 = a₁₁ + d = a₁₀ + 2d = a₉ + 3d = a₈ + 4d = 45 + 4d

69 = 45 + 4d

24 = 4d

d = 6

That also gives:

a₈ = 45

a₇ = 39

a₆ = 33

a₅ = 27

a₄ = 21

a₃ = 15

a₂ = 9

a₁ = 3

You could also calculate a₈ = 45 = a₁ + 7d to get a₁=3

The general formula for a term in an arithmetic sequence is a_n = a₁ + (n-1)d

The formula for this specific sequence is a_n = 3 + (n-1)(6)

That could be written as a_n = 6n -3

Check it out for n = 8 and for n = 12.

Answer 2

5n + (n-3)

n = 8 5 x 8 + (8 - 3) = 45

n= 12 5 x 12 + (12-3) = 69

n=1 5 x 1 +(1-3) = 3

n=2 5 x 2 + (2-3) = 9

n = 3 5 x 3 + (3-3) = 15

n = 4 5x 4 + (4-3) = 21

n=5 5 x 5 + ( 5-3) = 27 etc.

The terms differ by 6 as n increases by 1

Answer 3

The arithmetic sequence function is:

A(n) = A(1) + D*(n-1) where D is the common difference and A(1) is the 1st term

The common difference is D = [ a(12) - a(8)]/(12-8) = (69-45)/(12-8) = 24/4 = 6

A(k) = A(1)+ D(k-1)

A(8) = A(1) + 6(8-1)

45 = A1 + 6*7

45 = A1 + 42

A1 = 3

the function is A(n) = 3 + 6(n-1) = 3 + 6n-6 = 6n-3

A(1) = 3

A(2) = 9

A(3) = 15

etc....

A(8) = 45

etc...

A(12) = 69

The official answer: A(n) = 6n-3