Question

The 7th and 20th terms of an AP are 34 and 112 respectively .Find the Ap?

Answer 1

In an arithmetic progression (AP), the \(n\)th term can be expressed as:

\[
a_n = a + (n-1)d
\]

where:
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

Given:
- The 7th term \(a_7 = 34\)
- The 20th term \(a_{20} = 112\)

Using the formula for the \(n\)th term, we can write the following equations:

1. For the 7th term:
\[
a + 6d = 34 \quad \text{(1)}
\]

2. For the 20th term:
\[
a + 19d = 112 \quad \text{(2)}
\]

Now, we can solve these two equations simultaneously.

Subtract equation (1) from equation (2):

\[
(a + 19d) - (a + 6d) = 112 - 34
\]

This simplifies to:

\[
13d = 78
\]

Now, solve for \(d\):

\[
d = \frac{78}{13} = 6
\]

Now that we have \(d\), we can substitute it back into equation (1) to find \(a\):

\[
a + 6(6) = 34
\]
\[
a + 36 = 34
\]
\[
a = 34 - 36 = -2
\]

Now we have both the first term \(a\) and the common difference \(d\):

- First term \(a = -2\)
- Common difference \(d = 6\)

The general formula for the \(n\)th term of this AP can now be written as:

\[
a_n = -2 + (n-1) \cdot 6
\]

To summarize, the AP is:

\[
-2, 4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112, \ldots
\]