Sum of the First n Terms in an Arithmetic Progression with the nth Term Given

While working with arithmetic progressions (AP), it's essential to understand how to find the sum of the first n terms of the series when the nth term itself is provided. In this article, we will explore a specific example that showcases the process. Additionally, we will present the problem in both a detailed step-by-step manner and a more concise solution format.

Problem Statement

The nth term of an arithmetic progression is given by the formula Tn 3n - 15 / 2. Your task is to find the value of n for which the sum of the first n terms equals 84.

Solution: Step-by-Step Explanation

Step 1: Finding the First Term (T1)

First, we need to determine the first term of the arithmetic progression.

T1 3(1) - 15 / 2 -12 / 2 -6

Step 2: Determining the Common Difference (d)

Next, we will calculate the common difference using the first two terms. Here, we use the first and second terms to determine the common difference.

d T2 - T1

T2 3(2) - 15 / 2 -9 / 2 -4.5

d -4.5 - (-6) 1.5

Step 3: Using the Sum Formula for Arithmetic Progression

The sum of the first n terms of an arithmetic progression is found using the formula:

Sum of first n terms Sn n/2[2a (n-1)d]

Substituting the known values:

Sn n/2[2(-6) (n-1)(1.5)]

Sn n/2[-12 1.5n - 1.5]

Sn n/2[1.5n - 13.5]

Sn n(1.5n - 13.5) / 2

Step 4: Setting the Sum Equal to 84 and Solving for n

To find the value of n for which the sum of the first n terms is 84, we set the sum equal to 84 and solve for n.

n(1.5n - 13.5) / 2 84

Multiplying both sides by 2:

n(1.5n - 13.5) 168

Expanding and simplifying:

1.5n^2 - 13.5n - 168 0

To avoid decimals, multiply the entire equation by 2:

3n^2 - 27n - 336 0

Step 5: Solving the Quadratic Equation Using the Quadratic Formula

The quadratic formula n [-b ± sqrt(b^2 - 4ac)] / 2a is used to solve for n.

a 3, b -27, c -336

b^2 - 4ac (-27)^2 - 4(3)(-336) 729 4032 4761

sqrt(4761) 69

Substituting back into the formula:

n [27 ± 69] / 6

This gives us two potential solutions:

n (27 69) / 6 96 / 6 16

n (27 - 69) / 6 -42 / 6 -7

Since n must be a positive integer, the only valid solution is n 16.

Conclusion

Therefore, the value of n for which the sum of the first n terms is 84 is 16.

Key Takeaways

nth term formula: Tn 3n - 15 / 2 Sum of the first n terms formula: Sn n/2[2a (n-1)d] Understanding how to solve quadratic equations is key when dealing with more complex arithmetic progression problems.

Additional Resources

Come back to this article if you need a clearer understanding of arithmetic progressions and how to solve related mathematical problems. If you have further questions, explore more resources on Google or specific educational forums.