What is the common difference of this arithmetic sequence?

Question

In an arithmetic sequence, there are 94 terms, the 8th term is 63 and the sum of terms is 1000. What is the common difference?

Answer 1

Generally speaking, arithmetic sequences have the form:

 

a_n=a₁+(n-1)*d

 

Where:

 

a_n=the nth term in the sequence

a₁=the 1st term in the sequence

n=the number of terms in the sequence

d=the difference between consecutive terms (i.e. d=a_n-a_n-1=a₂-a₁=a₃-a₂, etc.)

 

The SUM of the terms of an arithmetic sequence, notated S_n, is given by:

 

S_n=[n/2]*[2a₁+(n-1)*d]

 

So if n=8 then:

 

a₈=a₁+7d=63

 

And if there are 94 terms in the sequence then:

 

S₉₄=[94/2]*[2a₁+93d]=47*(2a₁+93d)=94a₁+4371d=1000

 

So this is a system of equations where:

 

{     a₁ +         7d   =     63      (Multiply this first equation by -94 on both sides and add to second equation)

{  94a₁ +   4371d   =  1000

  -94a₁  -     658d   = -5922     (Add this new equation to second equation)

 

Therefore:

 

3713d = -4922

 

So:

 

d = -4922/3713                 (This fraction cannot be simplified any further)