Arithmetic sequences are a favorite topic on standardized tests like the SAT, ACT, SHSAT, and ISEE. Test-writers love them because they test your understanding of patterns, algebra, and critical thinking. These questions often look tricky but follow predictable rules—meaning that once you know the basics, they’re a perfect opportunity to score points.
Summing the first 100 integers
Imagine a lively math classroom in late 18th-century Germany. A young Carl Friedrich Gauss sits among his peers. The teacher, Herr Büttner, looking for a way to keep his students occupied, comes up with a challenging math task. He instructs the children to add up all the numbers from 1 to 100 — a task he expects will keep them quiet and busy for quite some time.
The students get to work, beginning with the slow and steady addition of 1+2+3+4+5… step by step. But there was a quicker way: Gauss, only 9 years old at the time, realized that if he paired the numbers from opposite ends of the range, each pair would add up to the same total:
- 1+100 = 101
- 2+99 = 101
- 3+98 = 101
And so on, all the way to 50+51=101.
Since each pair sums to 101, and there are 50 such pairs, Gauss quickly calculated:
50×101=5050
Within moments, Gauss had the answer. Büttner was stunned by Gauss’s quick response and the elegant method he used. Without formal instruction, Gauss had discovered a formula for the sum of an arithmetic series:
Sum =
n
2
× (first term + last term)
In this case, the first term is 1, the last term is 100, and there are 100 terms, so n = 100:
Sum =
100
2
× (1 + 100) = 50 × 101 = 5050
Note that this also works if we think of the first term as 0. The last term is still 100, we just have 101 terms now:
Sum =
101
2
× (0 + 100) = 50.5 × 100 = 5050
The sequence of integers from 1 to 100 is just one example of an arithmetic sequence, a type of sequence where each number is generated by adding a constant value, known as the common difference, to the previous number. In Gauss’s case, the sequence was simple: each number increased by 1. Arithmetic sequences can vary, depending on the starting term and the common difference.
Defining an Arithmetic Sequence
In general, an arithmetic sequence can be defined by:
- The first term, a1, which is the starting point of the sequence.
- The common difference, d, which is the amount added (or subtracted) to get from one term to the next.
The formula for the n-th term of an arithmetic sequence is:
an = a1 + (n − 1) × d
where:
- an is the n-th term,
- a1 is the first term,
- d is the common difference,
- n is the position of the term in the sequence.
Examples of Arithmetic Sequences
Here are some examples of arithmetic sequences, with the first term (a1) and the common difference (d) identified:
- Sequence: 2, 5, 8, 11
- First term, a1: 2
- Common difference, d: 3
- Each term increases by adding 3 to the previous term.
- Sequence: 10, 7, 4, 1
- First term, a1: 10
- Common difference, d: -3
- Each term decreases by subtracting 3 from the previous term.
- Sequence: −5, 0, 5, 10
- First term, a1: -5
- Common difference, d: 5
- Each term increases by adding 5.
Sum of an Arithmetic Sequence
As Gauss discovered, the sum of an arithmetic sequence can be calculated using a formula that simplifies the process of adding up each term individually. If you have an arithmetic sequence with n terms, the sum S of the sequence can be found with the formula:
Sum =
n
2
× (first term + last term)
Alternatively, if you know the first term (a1) and the common difference (d), you can find the last term using a1 + (n − 1)d, which, when substituted for an, becomes:
Sum =
n
2
× (2a1 + (n − 1)d)
Example of Finding the Sum
Consider the sequence 2,5,8,11,14… with 10 terms. To find the sum:
- Identify the values:
a1 = 2, d = 3, n = 10
- Calculate the last term, a10:
a10 = a1 + (10 − 1) × d = 2 + 9 × 3 = 2 + 27 = 29
- Apply the sum formula:
Sum =
10
2
× (2 + 29) = 5 × 31 = 155
Arithmetic sequences and their sums are powerful tools in mathematics, used in various fields like finance, physics, and everyday problem-solving.
