Solving an Arithmetic Progression Problem: First Ten Terms and Sum of First 20 Terms

In this article, we will explore a problem involving an arithmetic progression (AP), where the sum of the first ten terms is 50, and the fifth term is treble (three times) the second term. We will break down the problem into steps, using the properties of AP to find the first term and the common difference, and then calculate the sum of the first 20 terms.

Understanding Arithmetic Progression (AP)

An AP is a sequence of numbers where each term after the first is obtained by adding a fixed number (common difference) to the preceding term. The general form of an AP is:

Formula for the nth Term of an AP:

an a (n-1)d

Formula for the Sum of the First n Terms of an AP:

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

Given Information and Problem Setup

We are given the following information:

The sum of the first ten terms of the AP is 50. The fifth term is treble (three times) the second term.

Let the first term of the AP be a and the common difference be d.

Step-by-Step Solution

Step 1: Use the Sum of the First Ten Terms

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

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

For the first ten terms, n 10:

S_{10} frac{10}{2} [2a 9d] 50

Simplifying:

5 [2a 9d] 50

2a 9d 10 quad text{(Equation 1)}

Step 2: Use the Relationship Between the 5th and 2nd Terms

The 5th term of the AP is given by:

an a 4d

The 2nd term of the AP is given by:

a2 a d

According to the problem, the 5th term is treble the 2nd term:

a 4d 3 (a d)

Expanding the right side:

a 4d 3a 3d

Rearranging gives:

4d - 3d 3a - a

d 2a quad text{(Equation 2)}

Step 3: Substitute Equation 2 into Equation 1

Substituting d 2a into Equation 1:

2a 9(2a) 10

2a 18a 10

20a 10

a frac{10}{20} frac{1}{2}

Thus, the first term a is:

a frac{1}{2}

Using Equation 2 to find d:

d 2a 2 times frac{1}{2} 1

So, the common difference d is:

d 1

Step 4: Calculate the Sum of the First 20 Terms

Now we can find the sum of the first 20 terms S_{20}:

S_{20} frac{20}{2} [2a 19d]

S_{20} 10 [2 times frac{1}{2} 19 times 1]

S_{20} 10 [1 19] 10 times 20 200

Final Results

The first term is:

a frac{1}{2}

The common difference is:

d 1

The sum of the first 20 terms is:

S_{20} 200

Conclusion

By applying the properties of an arithmetic progression, we were able to find the first term, common difference, and the sum of the first 20 terms. This process demonstrates a systematic approach to solving complex arithmetic progression problems.

Related Content

Solving for the First Term and Common Difference in an AP

Understanding the Sum of Terms in an Arithmetic Sequence

Advanced Applications of Arithmetic Progression