How to calculate the number of terms for two APs.

How to calculate two APs with the same first term and last term but different common difference.

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? 

Solution

Since the first term and the last term of the two APs are the same

So let their equal first and last terms be x and y

Recall the nth term general formula of AP

Tn = a +(n-1)d

Solve for the first AP

Tn = y (last term)

a = x (First term)

n = 21 (This is the number of terms)

d = 9 (This is the common difference between terms) 

Substitute y, a, 21, and 9 for Tn, a, n, and d in the equation formula below

Tn = a +(n-1)d

y = x + (21-1)9

y = x +20 x 9

 y = x + 180   ---------- eqn(1) 

Solve for the Second AP

Tn = y (last term)

a = x (First term)

n = ?

d = 4 (This is the common difference between terms) 

Substitute y, a,  and 4 for Tn, a, and d in the equation formula below

Tn = a +(n-1)d

y = x + (n-1)4

y = x +4n - 4

 y = x + 4n - 4   ---------- eqn(2)

Equate equation 1 & 2

From equation 1 

 y = x + 180 

And from equation 2 

 y = x + 4n - 4

Therefore:

  x + 180  =  x + 4n - 4

Collect like terms

x - x +180+4 = 4n

0 + 184 = 4n

 4n = 184

Divide both sides by 4

4n/4 = 184/4

n = 48

The second AP has 48 terms.