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 called the common difference and is usually denoted by d.

• Each term can be found by adding d to the previous term.

Example:

• Here, the common difference d = 3 (5 − 2 = 3, 8 − 5 = 3, …)

• This is an arithmetic sequence.

Step 1: Identify the first term and common difference

• First term: usually written as a₁

• Common difference: d = a₂ − a₁

Example:

• a₁ = 7

• d = 10 − 7 = 3

Step 2: Find the nth term

The nth term formula for an arithmetic sequence is:

an=a1+(n−1)da_n = a_1 + (n – 1)dan​=a1​+(n−1)d

Where:

• a_n = nth term

• a₁ = first term

• d = common difference

• n = term number

Example:

Find the 10th term:

a10=2+(10−1)(3)=2+9(3)=2+27=29a_{10} = 2 + (10 – 1)(3) = 2 + 9(3) = 2 + 27 = 29a10​=2+(10−1)(3)=2+9(3)=2+27=29

✅ 10th term = 29

Step 3: Find the sum of n terms

The sum of the first n terms of an arithmetic sequence is:

Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)Sn​=2n​(a1​+an​)

or

Sn=n2[2a1+(n−1)d]S_n = \frac{n}{2}[2a_1 + (n-1)d]Sn​=2n​[2a1​+(n−1)d]

Example:

Step 1: Find a₁₀ (already done): 29

Step 2: Sum formula:

S10=102(2+29)=5(31)=155S_{10} = \frac{10}{2}(2 + 29) = 5(31) = 155S10​=210​(2+29)=5(31)=155

✅ Sum of first 10 terms = 155

Step 4: Write a general arithmetic sequence

• General term formula: a_n = a₁ + (n-1)d

• Sequence example: a₁ = 4, d = 5

✅ Sequence: 4, 9, 14, 19, 24, …

Step 5: Real-life examples

Common beginner mistakes

1. ❌ Forgetting to subtract terms to find d

2. ❌ Using the wrong formula for nth term or sum

3. ❌ Confusing the term number n with the value of the term

4. ❌ Forgetting the first term in calculations

Summary

• Arithmetic sequence = constant difference d

• nth term: a_n = a₁ + (n-1)d

• Sum of first n terms: S_n = n/2(a₁ + a_n) or S_n = n/2[2a₁ + (n-1)d]

• Useful in real-world scenarios like payments, seating, or growth