Solving Arithmetic Progression Problems: A Detailed Example

Arithmetic Progression (AP) is a fundamental concept in mathematics, used extensively in various applications. This article will walk you through a detailed problem-solving process to find the sum of the third and fourth terms of an AP, given the sum of the series and the value of its sixth term. We will use both theoretical explanations and step-by-step calculations to illustrate the solution.

Understanding the Problem

The problem at hand is to find the sum of the third and fourth terms of an arithmetic progression (AP) given that the sum of the AP is 99 and the sixth term is 39. The key formulas and steps involved are:

The sum of the first n terms of an AP is given by:

S_n frac{n}{2} times (2a (n-1)d)

The nth term of an AP is given by:

a_n a (n-1)d

Step-by-Step Solution

Step 1: Setting Up the Equations

Given information:

The sum of the AP: 99 The 6th term: 39

We need to determine the following:

Sum of the first n terms equation:

S_n frac{n}{2} times (2a (n-1)d) 99

6th term equation:

a_6 a 5d 39

Step 2: Finding the Sum of the 3rd and 4th Terms

The 3rd and 4th terms can be expressed as:

3rd term a3:

a_3 a 2d

4th term a4:

a_4 a 3d

The sum of the 3rd and 4th terms is:

a_3 a_4 (a 2d) (a 3d) 2a 5d

Step 3: Relating 2a 5d to Known Values

From the 6th term equation:

a 5d 39

Multiplying both sides by 2:

2a 10d 78

We need to express 2a 5d in terms of the known values:

2a 5d 2a 5d - 5d 5d 2a 10d - 5d 2a 10d - 5d 78 - 5d

Step 4: Assuming n and Solving for d

Assuming n 9 (a common choice for AP problems), we can substitute in the sum formula:

S_9 frac{9}{2} times (2a 8d) 99

This simplifies to:

frac{9}{2} times (2a 8d) 99

9(a 4d) 99

a 4d 11

Now we have two equations:

S9 equation:

a 4d 11

6th term equation:

a 5d 39

Step 5: Solving the System of Equations

Subtract the first equation from the second:

(a 5d) - (a 4d) 39 - 11

d 28

Substitute d 28 into the first equation:

a 4 times 28 11

a 112 11

a 11 - 112

a -101

Step 6: Finding the 3rd Term

The 3rd term a3 is:

a_3 a 2d -101 2 times 28 -101 56 -45

Step 7: Finding the Sum of the 3rd and 4th Terms