135 LearnersLast updated on August 5th, 2025
Math Formulas for Sequences and Series
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.
List of Math Formulas for 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.
Math Formula for Arithmetic Sequences
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.
Math Formula for Arithmetic Series
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.
Math Formula for Geometric Sequences
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.
Math Formula for Geometric Series
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.
Tips and Tricks to Memorize Sequences and Series Math Formulas
Students often find math formulas tricky and confusing.
Here are some tips and tricks to master sequences and series formulas:
- Use simple mnemonics like "Arithmetic Adds" and "Geometric Grows."
- Connect the use of sequences and series with real-life examples, such as savings plans (series) or patterns (sequences).
- Use flashcards to memorize the formulas and rewrite them for quick recall.
- Create a formula chart for quick reference.
Common Mistakes and How to Avoid Them While Using Sequences and Series Math Formulas
Students often make errors when calculating sequences and series. Here are some mistakes and ways to avoid them to master these concepts.
Mistake 1
Forgetting to Apply the Correct Formula for the Sequence
Students sometimes use the wrong formula for the type of sequence they are working with.
To avoid this, ensure you identify whether the sequence is arithmetic or geometric before applying the formula.
Mistake 2
Calculation Errors in Series Formulas
When calculating the sum of series, students may make calculation errors.
To avoid these, double-check the values and verify whether the calculations are correct.
Mistake 3
Confusing Common Difference and Common Ratio
Students sometimes confuse the common difference in arithmetic sequences with the common ratio in geometric sequences.
Remember: arithmetic sequences have a difference, geometric ones have a ratio.
Mistake 4
Overlooking the First Term in Formulas
Students often forget to include the first term in their calculations.
Always verify that the first term is considered when using the sequence or series formulas.
Mistake 5
Mistaking the Number of Terms
Students sometimes miscount the number of terms in a sequence or series, leading to errors.
Always count the terms carefully and apply the correct formula.
Examples of Problems Using Sequences and Series Math Formulas
Problem 1
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.
Explanation
Using the formula ( a_n = a_1 + (n - 1) cdot d ):
a_{10} = 3 + (10 - 1) cdot 5
= 3 + 45
= 48
Problem 2
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.
Explanation
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
Problem 3
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.
Explanation
Using the formula ( a_n = a_1 cdot r{(n-1)} ):
a_6 = 2 cdot 3{(6-1)}
= 2 cdot 243
= 486
Problem 4
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.
Explanation
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
Problem 5
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.
Explanation
Using the formula ( a_n = a_1 + (n - 1) cdot d ):
a_7 = 10 + (7 - 1) cdot 4
= 10 + 24
= 34 .
FAQs on Sequences and Series Math Formulas
1.What is the formula for an arithmetic sequence?
2.What is the formula for the sum of an arithmetic series?
3.What is the formula for a geometric sequence?
4.What is the formula for the sum of a geometric series?
5.How do I identify if a sequence is arithmetic or geometric?
Glossary for Sequences and Series Math Formulas
- Arithmetic Sequence: A sequence in which each term is derived by adding a constant to the previous term.
- Geometric Sequence: A sequence in which each term is derived by multiplying the previous term by a constant.
- Arithmetic Series: The sum of the terms in an arithmetic sequence.
- Geometric Series: The sum of the terms in a geometric sequence.
- Common Ratio: The fixed number that each term in a geometric sequence is multiplied by to get the next term.
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Jaskaran Singh Saluja
About the Author
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
