Question

The 7th and 20th term of an are 34 and 113 respectively find the ap

Answer 1

To find the arithmetic progression (AP) given the 7th term \( a_7 = 34 \) and the 20th term \( a_{20} = 113 \), we can use the formula for the \( n \)-th term of an AP, which is given by:

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

where \( a \) is the first term and \( d \) is the common difference.

1. For the 7th term:
\[
a_7 = a + 6d = 34 \quad \text{(Equation 1)}
\]

2. For the 20th term:
\[
a_{20} = a + 19d = 113 \quad \text{(Equation 2)}
\]

Now, we can solve these equations step-by-step.

### Step 1: Subtract Equation 1 from Equation 2

\[
(a + 19d) - (a + 6d) = 113 - 34
\]

This simplifies to:

\[
19d - 6d = 79
\]
\[
13d = 79
\]
\[
d = \frac{79}{13} = 6.0769 \quad \text{(approximately)}
\]

### Step 2: Substitute \( d \) back into Equation 1 to find \( a \)

Substituting \( d \) into Equation 1:

\[
a + 6(6.0769) = 34
\]

Calculating \( 6 \times 6.0769 \):

\[
6 \times 6.0769 \approx 36.4614
\]
\[
a + 36.4614 = 34
\]
\[
a = 34 - 36.4614
\]
\[
a \approx -2.4614
\]

### Final Form

The values of \( a \) and \( d \) are:

\[
a \approx -2.4614, \quad d \approx 6.0769
\]

### General Term of the AP

The general term \( a_n \) can be expressed as:

\[
a_n = -2.4614 + (n-1) \cdot 6.0769
\]

Thus, the arithmetic progression is determined, and the first few terms can be calculated based on these values.