Sequence and series

Question

two APs have the same first and last terms. The first AP has 21 terms with a common difference of 9. How many terms has the other AP if its common difference is 4?

Answer 1

An Arithmetic Progression has a common difference between terms.  Thus, the formula for the n^th term is:

      n^th term = first term + (n-1)d               [where d is the common difference]

          a_n  =    a₁ + (n-1)d

 

For the AP with 21 terms and d=9, the 21st term is:

         a₂₁ = a₁ + (21-1)*9

 

For the AP with 9 terms and d=4, the xth tem is:

         b_x = b₁ + (x-1)*4

 

Now, consider same first (a₁=b₁) and last (a₂₁=b_x) terms:

    a₁ + (21-1)*9 = a₁ + (x-1)*4

            180 = 4x - 4

             184 = 4x

               46 = x

 

The second Arithmetic Progression has 46 terms.