Question

If 5th and 9th terms of an A.P. are 43 and 79 respectively, find the A.P.

Hint:

A quick way to find the common difference \(d\) when given two terms \(a_m\) and \(a_n\) is to use the formula \(d = \frac{a_m - a_n}{m - n}\). Here, \(d = \frac{79 - 43}{9 - 5} = \frac{36}{4} = 9\).

Answer 1

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is determined by its first term (\(a\)) and common difference (\(d\)). We are given two terms of the A.P., which allows us to set up a system of two linear equations in \(a\) and \(d\). Solving this system will give us the required values.

Step 2: Key Formula or Approach:
The formula for the n-th term of an A.P. is \(a_n = a + (n-1)d\).

Step 3: Detailed Explanation:
Given: The 5th term is 43: \(a_5 = a + (5-1)d = a + 4d = 43\) ---(Equation 1)
The 9th term is 79: \(a_9 = a + (9-1)d = a + 8d = 79\) ---(Equation 2)
To solve this system, we can subtract Equation 1 from Equation 2: \[ (a + 8d) - (a + 4d) = 79 - 43 \] \[ 4d = 36 \] \[ d = \frac{36}{4} = 9 \] Now that we have the common difference (\(d=9\)), substitute it back into Equation 1 to find the first term (\(a\)): \[ a + 4(9) = 43 \] \[ a + 36 = 43 \] \[ a = 43 - 36 = 7 \] The first term is \(a = 7\) and the common difference is \(d = 9\).
The A.P. is formed by starting with \(a\) and repeatedly adding \(d\). The A.P. is: 7, (7+9), (16+9), ...
The A.P. is: 7, 16, 25, 34, ...

Step 4: Final Answer:
The required Arithmetic Progression is 7, 16, 25, 34, ...