The first term of an Arithmetic progression is 3 and the fifth terms 9.find the number of terms in the progression of the last term is 81
The general formula for an arithmetic progression is given by:
\[a_n = a_1 + (n-1)d\]
where \(a_n\) is the last term, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.
Given that the first term \(a_1\) is 3 and the fifth term is 9, we can find the common difference:
\[9 = 3 + 4d\]
Simplifying, we get:
\[4d = 6\]
\[d = \dfrac{6}{4}\]
\[d = 1.5\]
Now, we need to find the number of terms \(n\) when the last term \(a_n\) is 81. Using the formula:
\[81 = 3 + (n-1)(1.5)\]
Simplifying, we get:
\[78 = (n-1)(1.5)\]
Dividing both sides by 1.5, we get:
\[52 = n-1\]
\[n = 53\]
Therefore, there are 53 terms in the arithmetic progression.