Result:

1. Nth Term Formula (Explicit Formula)

The formula to find any term in an arithmetic sequence:

Where: aₙ = nth term, a₁ = first term, n = term position, d = common difference

Example: Find the 20th term of sequence 5, 9, 13, 17... → a₂₀ = 5 + (20-1)×4 = 5 + 76 = 81

2. Sum of Arithmetic Series Formula

Two equivalent formulas for calculating the sum of n terms:

Where: Sₙ = sum of n terms, n = number of terms, a₁ = first term, aₙ = last term, d = common difference

Example: Sum of first 10 terms of 2, 5, 8, 11... → S₁₀ = 10/2 × [2(2) + (10-1)×3] = 5 × [4 + 27] = 155

3. Common Difference Formula

Calculate the constant difference between consecutive terms:

Example: For sequence 7, 12, 17, 22... → d = 12 - 7 = 5

4. Recursive Formula for Arithmetic Sequence

Express each term using the previous term:

Example: If a₁ = 3 and d = 4, then a₂ = a₁ + 4 = 7, a₃ = a₂ + 4 = 11, etc.

Step 1: Select Your Calculation Type

Choose what you need to find from the dropdown menu: nth term, sum of series, common difference, first term, or number of terms.

Step 2: Enter Known Values

Input the values you know. The calculator will show only the relevant fields based on your selected calculation type.

Step 3: Calculate and Review

Click the Calculate button to get instant results with detailed step-by-step solutions showing exactly how the answer was derived using arithmetic sequence formulas.

Example 1: Finding the 52nd Term

Problem: Find the 52nd term of the arithmetic sequence: 3, 8, 13, 18, 23...

Solution:

First term (a₁) = 3, Common difference (d) = 8 - 3 = 5

Using formula: aₙ = a₁ + (n - 1)d

a₅₂ = 3 + (52 - 1) × 5 = 3 + 51 × 5 = 3 + 255 = 258

Example 2: Sum of Arithmetic Series

Problem: Calculate the sum of the first 30 terms: 4, 7, 10, 13, 16...

Solution:

First term (a₁) = 4, Common difference (d) = 3, Number of terms (n) = 30

Using formula: Sₙ = n/2 × [2a₁ + (n-1)d]

S₃₀ = 30/2 × [2(4) + (30-1)×3] = 15 × [8 + 87] = 15 × 95 = 1425

Example 3: Finding Common Difference

Problem: In an arithmetic sequence, a₁ = 12 and a₁₅ = 84. Find the common difference.

Solution:

Using: aₙ = a₁ + (n - 1)d

84 = 12 + (15 - 1)d

84 = 12 + 14d → 72 = 14d → d = 5.14

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Construction & Architecture

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Time Management

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Physics & Science

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What is an arithmetic sequence?

An arithmetic sequence (or arithmetic progression) is a sequence of numbers where the difference between consecutive terms is constant. This fixed difference is called the common difference (d). For example: 2, 5, 8, 11, 14... has a common difference of 3.

How do you find the nth term of an arithmetic sequence?

Use the formula: aₙ = a₁ + (n - 1)d, where aₙ is the nth term, a₁ is the first term, n is the term position, and d is the common difference. For example, to find the 10th term of sequence 3, 7, 11, 15...: a₁₀ = 3 + (10-1)×4 = 39.

What is the formula for the sum of an arithmetic series?

There are two formulas: Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2(a₁ + aₙ), where Sₙ is the sum, n is the number of terms, a or a₁ is the first term, d is the common difference, and aₙ is the last term.

How do you calculate the common difference in an arithmetic sequence?

Subtract any term from the term that follows it: d = aₙ - aₙ₋₁. For example, in the sequence 5, 9, 13, 17..., the common difference d = 9 - 5 = 4.

What is the recursive formula for an arithmetic sequence?

The recursive formula is aₙ = aₙ₋₁ + d, where each term is found by adding the common difference to the previous term. This requires knowing the first term a₁ and the common difference d.

Can an arithmetic sequence have a negative common difference?

Yes, when the common difference is negative, the sequence decreases. For example: 20, 15, 10, 5, 0... has d = -5. The same formulas apply regardless of whether d is positive or negative.

What's the difference between arithmetic and geometric sequences?

Arithmetic sequences have a constant difference between terms (addition/subtraction), while geometric sequences have a constant ratio (multiplication/division). For example, 2, 4, 6, 8 is arithmetic (difference of 2), while 2, 4, 8, 16 is geometric (ratio of 2).

How do I find the number of terms in an arithmetic sequence?

Rearrange the nth term formula: n = [(aₙ - a₁)/d] + 1. For example, in sequence 5, 8, 11, 14..., to find how many terms until we reach 50: n = [(50-5)/3] + 1 = 16 terms.

1. Always Identify the Pattern First

Calculate the difference between consecutive terms to confirm you have an arithmetic sequence and find the common difference.

2. Write Down What You Know

List the given values (first term, common difference, which term, etc.) to determine which formula to use.

3. Choose the Right Formula

For finding a term use aₙ = a₁ + (n-1)d. For sums, use Sₙ = n/2[2a₁ + (n-1)d] or Sₙ = n/2(a₁ + aₙ).

4. Check Your Answer

Verify by calculating a few terms manually or using the calculator to ensure your solution makes sense.

5. Handle Negative Differences Carefully

When the sequence decreases, the common difference is negative. Make sure to use negative values correctly in formulas.