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 it's common difference is 4?

Answer 1

The formula for the nth term of an arithmetic progression is given by:

An = A1 + (n-1)d,

where An is the nth term, A1 is the first term, n is the number of terms, and d is the common difference.

For the first AP:

A1 = first term = A21 = last term
n = 21
d1 = 9

Using the formula, we can find A1:

A1 = A21 - (21-1)d1
A1 = A21 - 20d1

For the second AP:

d2 = 4

Using the formula, we can find the number of terms (n2) in the second AP:

A1 = A2 - (2-1)d2
A1 = A2 - d2

Since A1 is equal to the last term of the first AP (A1 = A21), we can substitute A21 - 20d1 for A1:

A21 - 20d1 = A2 - d2

Simplifying, we get:

21 - 20(9) = A2 - 4
21 - 180 = A2 - 4
A2 - 4 = -159

Thus, the number of terms in the second AP is 159.