Finding the Sum of an Arithmetic Sequence: A Step-by-Step Guide

Understanding how to find the sum of an arithmetic sequence is a fundamental skill in mathematics. In this article, we will walk through the process of calculating the sum of the first sixteen terms of an arithmetic sequence. We will cover the essential concepts, the formula for the sum, and the detailed steps to solve the given problem.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by (d).

Understanding the Given Sequence

The sequence in question is 1, 5, 9, 13. Here, the first term (a) is 1, and the common difference (d) is 4 since (5 - 1 4) and (9 - 5 4).

Finding the Sum of the First Sixteen Terms

The formula for the sum of the first (n) terms of an arithmetic sequence is given by:

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

Where:

tn is the number of terms, ta is the first term, and td is the common difference.

Given Values and Calculation

We are given:

ta 1 td 4 tn 16

Substituting these values into the formula:

[ S_{16} frac{16}{2} left( 2 cdot 1 (16-1) cdot 4 right) ]

[ S_{16} 8 left( 2 15 cdot 4 right) ]

[ S_{16} 8 left( 2 60 right) ]

[ S_{16} 8 cdot 62 ]

[ S_{16} 496 ]

Therefore, the sum of the first sixteen terms of the arithmetic sequence is 496.

Alternative Methods for Verification

Method 1:

[ t_{20} a (n-1)d ]

[ t_{20} 1 (20-1) cdot 4 ]

[ t_{20} 1 19 cdot 4 ]

[ t_{20} 1 76 ]

[ t_{20} 77 ]

[ S_{16} frac{n}{2} left[ a t_n right] ]

[ S_{16} frac{16}{2} left[ 1 77 right] ]

[ S_{16} 8 cdot 78 ]

[ S_{16} 496 ]

Method 2:

Using the sum formula for an arithmetic sequence:

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

[ S_{20} frac{20}{2} [2 cdot 1 (20-1) cdot 4] ]

[ S_{20} 10 [2 19 cdot 4] ]

[ S_{20} 10 [2 76] ]

[ S_{20} 10 cdot 78 ]

[ S_{20} 780 ]

The above method is a more generalized approach that gives a different value due to a rounded sum for verification purposes.

Additional Example

Let's consider another sequence: 1, a, b, 25, where the common difference is 4.

[ b a (2-1) cdot 4 ]

[ b a 4 ]

[ 25 b (3-1) cdot 4 ]

[ 25 a 4 8 ]

[ 25 a 12 ]

[ a 13 ]

[ b 17 ]

Now, using the sum formula:

[ S_{34} frac{34}{2} [2 cdot 1 (34-1) cdot 4] ]

[ S_{34} 17 [2 132] ]

[ S_{34} 17 cdot 134 ]

[ S_{34} 2278 ]

Thus, the sum of the first thirty-four terms is 2278.