How to find the nth term of an Arithmetic Sequence?

Answer – Use the formula: a_n = a₁ + (n – 1)d

Where:
a_n = nth term,
a = first term,
d = common difference,
n = term number.

Substitute the values of a, d, and n into the formula to calculate a_n.

Steps to find the n^th Term of an Arithmetic Sequence

Step 1: Identify the First and Second Term:

• 1st and 2nd term, that is a₁ and a₂.

Step 2: Find the Common Difference:

• Subtract the first term from the second term (d = a₂ – a₁)

Step 3: Use the Formula:

a_n = a₁ + (n – 1)d

Step 4: Substitute Values:

• Plug in the values of a, d, and n into the formula.

Step 5: Simplify and Solve:

• Perform the arithmetic operations to find the nth term.

Solved Problems – How to find the nth term of an Arithmetic Sequence?

Question 1: Find the 9th term of the given series, {1, 4, 7, 10, 13, 16,….}

Solution:

Term number (n) is 9,

1st-term, a₁ = 1 
2nd term, a₂ = 4,

Now find the common difference, 
d = a₂ – a₁ = 4 – 1 = 3

Now the 9th term, 

a₉ = First term + (Last term – 1) × common difference 
= a₁ + (n – 1)d
= 1 + (9 – 1) × 3
= 1 + 8 × 3
= 1 + 24
= 25 

So, the 9th term is 25.

Question 2: Find the 7th term of an AP whose 3rd term is 9 and 5th term is 15?

Solution:

Given

• 3rd term (a₃) = 9
• 5th term (a₅ ) = 15

Here we have to find common difference and first term(a₁).

a₃ = a₁ + 2d = 9  [from the formula] ⇢  (1)

And, a₅ = a₁ + 4d = 15  ⇢  (2)

Solve (1) and (2),

a₁ + 2d = 9 ⇢  (1)
a₁ + 4d = 15  ⇢  (2)

Let’s apply subtraction between (1) and (2) 
2d = 6, d=3

So, the common difference is 3

Now put the value of d in any one equation, here put the value of d in (1)
a₁ + 2d = 9
= a₁ + 2 × 3 = 9    [d = 3]
= a₁ = 9 – 6
= a₁ = 6

So the first term is 3

Now to find the 7th term,

Apply the formula for finding the nth term, here n = 7
a₇ = First term + (7th term – 1) × common difference
a₇ = a₁ + (7 – 1)d
a₇ = 3 + 6 × 3   [d = 3 and a₁ = 3]
a₇ = 21

So the 7th term is 21.

Related Articles:

• What are the 4 types of sequences?
• Arithmetic Series
• Common difference
• What is the nth term?