Last updated on August 2nd, 2025
In mathematics, sequences and series are fundamental concepts involving ordered lists of numbers and their sums. Sequences can be arithmetic, geometric, or more complex, while series are the sums of sequences. In this topic, we will learn the formulas for various types of sequences and series.
Sequences and series are key concepts in mathematics. Let’s learn the formulas to calculate the terms and sums of arithmetic and geometric sequences and series.
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
It is calculated using the formula: [ a_n = a_1 + (n - 1) * d ] where ( a_n ) is the nth term, ( a_1 ) is the first term, \( d \) is the common difference, and \( n \) is the term number.
The sum of the first n terms of an arithmetic sequence is called an arithmetic series.
The formula is: S_n = n/2 (a_1 + a_n) (or) S_n = n/2 (2a + (n - 1) d) where ( S_n ) is the sum of the first n terms, ( a_1 ) is the first term, ( a_n ) is the nth term, and ( d ) is the common difference.
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
The formula is: [ a_n = a_1 * r(n-1) ] where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.
The sum of the first n terms of a geometric sequence is called a geometric series.
The formula is: (S_{n} = a_{1}\cdot frac{1-r{n}}{1-r}), where ( S_n ) is the sum of the first n terms, ( a_1) is the first term, and ( r ) is the common ratio.
Students often find math formulas tricky and confusing.
Here are some tips and tricks to master sequences and series formulas:
Students often make errors when calculating sequences and series. Here are some mistakes and ways to avoid them to master these concepts.
Find the 10th term of the arithmetic sequence where the first term is 3 and the common difference is 5.
The 10th term is 48.
Using the formula ( a_n = a_1 + (n - 1) cdot d ):
a_{10} = 3 + (10 - 1) cdot 5
= 3 + 45
= 48
Calculate the sum of the first 8 terms of the arithmetic sequence where the first term is 4 and the common difference is 3.
The sum is 100.
Using the formula ( S_n = frac{n}{2} cdot (2a_1 + (n - 1) cdot d) ):
S_8 = frac{8}{2} cdot (2 cdot 4 + (8 - 1) cdot 3)
= 4 cdot (8 + 21)
= 4 cdot 29
= 116
Find the 6th term of the geometric sequence where the first term is 2 and the common ratio is 3.
The 6th term is 486.
Using the formula ( a_n = a_1 cdot r{(n-1)} ):
a_6 = 2 cdot 3{(6-1)}
= 2 cdot 243
= 486
Calculate the sum of the first 5 terms of the geometric sequence where the first term is 1 and the common ratio is 2.
The sum is 31.
Using the formula ( S_n = a_1 cdot frac{1 - rn}{1 - r} ):
S_5 = 1 cdot frac{1 - 25}{1 - 2}
= frac{1 - 32}{-1}
= 31
What is the 7th term of the arithmetic sequence where the first term is 10 and the common difference is 4?
The 7th term is 34.
Using the formula ( a_n = a_1 + (n - 1) cdot d ):
a_7 = 10 + (7 - 1) cdot 4
= 10 + 24
= 34 .
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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