How to Find the Sum of the First 100 Terms of an Arithmetic Sequence

Have you ever struggled to find the sum of the first 100 terms of an arithmetic sequence? If you are given just a couple of terms, how can you proceed? In this guide, we will walk you through the process of finding the sum of the first 100 terms of an arithmetic sequence where the second term is 20 and the fifth term is 11. This step-by-step approach will help you solve similar problems efficiently.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference (denoted by d), to the previous term. The general formula for the nth term of an arithmetic sequence is:

an a1 (n-1)d

Given Information

The second term, a2 20 The fifth term, a5 11

To find the sum of the first 100 terms, we first need to determine the first term and the common difference of the sequence. With the given terms, we can set up and solve a system of equations.

Step-by-Step Solution

Step 1: Set Up the Equations

Given the information, we can write down the following equations:

a2 a1 d 20a5 a1 4d 11

Step 2: Solve the Equations

To solve these equations, we can use substitution or elimination methods. Let's use the elimination method:

From the first equation: d 20 - a1

Substituting d 20 - a1 into the second equation:

a1 4(20 - a1) 11

Simplifying this equation:

a1 80 - 4a1 11

-3a1 -69

a1 23

Step 3: Find the First Term and Common Difference

Now that we have the first term a1 23, we can find the common difference:

d 20 - a1 20 - 23 -3

Step 4: Calculate the Sum of the First 100 Terms

The sum of the first n terms of an arithmetic sequence can be calculated using the formula:

Sn frac{n}{2} [2a1 (n-1)d]

For n 100:

S100 frac{100}{2} [2(23) (100-1)(-3)]

S100 50 [46 99(-3)]

S100 50 [46 - 297]

S100 50[-251]

S100 -12550

Final Result

The sum of the first 100 terms of the arithmetic sequence is -12550.

Conclusion

By following the steps outlined above, we can systematically solve for the sum of the first 100 terms of an arithmetic sequence when given two specific terms. This method can be applied to similar problems involving arithmetic sequences, providing a clear and concise approach to solving such questions.