Finding the First Term in an Arithmetic Sequence: A Comprehensive Guide

When dealing with arithmetic sequences, it's important to understand how to find the first term given specific values such as the common difference, the last term, and the sum of all the terms. This guide will walk you through the steps using a practical example, ensuring you can apply the same principles to similar problems.

Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between any two successive members is constant. This difference is known as the common difference. The formula for the nth term and the sum of the terms in an arithmetic sequence are fundamental for solving many problems.

Key Formulas

The formula for the sum of an arithmetic series is:

Sn frac{n}{2} (a l)

where:

Sn - the sum of the series n - the number of terms in the series a - the first term l - the last term

The formula for the last term of an arithmetic sequence is:

l a (n-1) d

where:

d - the common difference l - the last term a - the first term n - the number of terms in the series

Solving for the First Term

Consider the scenario where the common difference is 4, the last term is 72, and the sum of all terms is 600. We need to find the first term.

Step-by-Step Solution

Set up the equations based on the given information:

72 a (n-1)4 600 frac{n}{2}(a 72)

Solve the equations step by step:

From the last term equation:

a 72 - 4n 4

Substitute into the sum equation:

600 frac{n}{2}(72 - 4n 4 72)

Rearrange and simplify:

600 frac{n}{2}(148 - 4n)

600 74n - 2n^2

2n^2 - 74n 600 0

n^2 - 37n 300 0

Use the quadratic formula to solve for n:

n frac{-b pm sqrt{b^2 - 4ac}}{2a}

n frac{37 pm sqrt{1369 - 1200}}{2}

n frac{37 pm 13}{2}

n frac{50}{2} 25

or

n frac{24}{2} 12

Find the corresponding first term for each value of n:

For n 25:

a 72 - 4 cdot 25

a 72 - 100

a -24

For n 12:

a 72 - 4 cdot 12

a 72 - 48

a 28

Verify both solutions:

For n 25, a -24:

Correct last term: -24 24 cdot 4 72 Correct sum: frac{25}{2} (-24 72) 600

For n 12, a 28:

Correct last term: 28 11 cdot 4 72 Correct sum: frac{12}{2} (28 72) 600

Conclusion:

Both solutions are valid, so the first term can be either -24 with n 25, or 28 with n 12.

Finding the First Term: Summary

You can apply the methods used here to solve similar problems involving arithmetic sequences. Understanding the formulas and solving the equations step by step ensure accurate results. The key is to set up the correct equations and use the quadratic formula when necessary.

Additional Resources

For further reading and practice, consider exploring resources on algebra and arithmetic sequences. Online platforms such as Khan Academy,Purplemath, and Mathway can provide additional examples and practice problems.

Keywords: arithmetic sequence, first term, common difference, sum of terms