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

1 answer

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.