Write each sum using summation notation.

Question

Find the first term of an arithmetic progression if a₇=35 and a₁₂=60.

Answer 1

The equation of an arithmetic sequence is: a_n=a₁ + (n-1)d, where d = common difference.  We have 2 variables here, a₁ and d, so we're going to set up a system of equations to solve them:

 

a₇ = a₁ + (7-1)d   -->  35 = a₁ + 6d

a_{12 =} a₁ + (12-1)d --> 60 =  a₁ + 11d

 

If you take equation 2 and subtract equation 1, you can eliminate the a₁ term:

       60 = a₁ + 11d

  -   (35 = a₁ + 6d )

 

25 = 5d  

d = 5

 

Now, substitute d = 5 into either of the 2 equations; I'll use equation 1:

35 = a₁ + 6(5)

35 = a₁ + 30

a₁ = 5.

 

Therefore, the first term of the sequence is 5.