Physics and Mathematics
nth term of A.P.
1. Introduction
In an arithmetic progression, it is often required to find:
- the 5ᵗʰ term
- the 10ᵗʰ term
- or the [n^{th}] term (general term)
Instead of writing all the terms, we use a direct formula.
2. General Form of an AP
[a, a + d, a + 2d, a + 3d, …]
Where:
- [a] = first term
- [d] = common difference
3. Derivation of the [n^{th}] Term Formula
Let:
- First term = [a]
- Common difference = [d]
Then:
- 1ˢᵗ term = [a]
- 2ⁿᵈ term = [a + d]
- 3ʳᵈ term = [a + 2d]
So, the [n^{th}] term is:
[ aₙ = a + (n − 1)d ]
4. Formula for [n^{th}] Term
[n^{th}] term of an AP:
[ aₙ = a + (n − 1)d ]
Where:
- [aₙ] = [n^{th}] term
- [a] = first term
- [d] = common difference
5. Examples with Solutions
Example 1.
Find the 10ᵗʰ term of the AP: [2, 5, 8, 11, …]
Solution:
a = 2, d = 3, n = 10
[ a₁₀ = 2 + (10 − 1) × 3 ]
[ a₁₀ = 2 + 27 = 29 ]
Answer:
The 10ᵗʰ term is 29.
Example 2.
Find the 15ᵗʰ term of the AP: [7, 4, 1, −2, …]
Solution:
a = 7, d = −3, n = 15
[ a₁₅ = 7 + (15 − 1)(−3) ]
[ a₁₅ = 7 − 42 = −35 ]
Answer:
The 15ᵗʰ term is −35.
Example 3.
Find the [n^{th}] term of the AP whose first term is [5] and common difference is [4].
Solution:
a = 5, d = 4
[ aₙ = 5 + (n − 1) × 4 ]
[ aₙ = 5 + 4n − 4 ]
[ aₙ = 4n + 1 ]
Answer:
The nᵗʰ term is [4n + 1].
Example 4.
Find the 20ᵗʰ term of the AP: [−3, −1, 1, 3, …]
Solution:
a = −3, d = 2, n = 20
[ a₂₀ = −3 + (20 − 1) × 2 ]
[ a₂₀ = −3 + 38 = 35 ]
Answer:
The 20ᵗʰ term is 35.
Example 5.
Find the 7ᵗʰ term of the AP: [1/2, 3/2, 5/2, …]
Solution:
a = 1/2, d = 1, n = 7
[ a₇ = 1/2 + (7 − 1) × 1 ]
[ a₇ = 1/2 + 6 ]
[ a₇ = 13/2 ]
Answer:
The 7ᵗʰ term is 13/2.
Important Observations
- If [d > 0], the AP is increasing.
- If [d < 0], the AP is decreasing.
- If [d = 0], all terms are equal.
- The formula works for any real value of [a] and [d].
6. Conceptual Questions with Solutions
1. What is meant by the [n^{th}]term of an AP?
The [n^{th}] term is the general term that represents the value of the sequence at position n.
2. What is the formula for the [n^{th}] term of an AP?
The formula is [aₙ = a + (n − 1)d].
3. What does the symbol a represent?
a represents the first term of the arithmetic progression.
4. What does d represent?
d represents the common difference between consecutive terms.
5. Why is [(n − 1)] used in the formula?
Because the first term corresponds to [n = 1], so the increase happens [(n − 1)] times.
6. Can the [n^{th}] term be negative?
Yes, if the AP is decreasing, the nᵗʰ term can be negative.
7. Can n be zero?
No. n always starts from 1 because sequences are indexed from the first term.
8. What happens if d = 0?
All terms become equal, and [aₙ = a].
9. Is the [n^{th}] term formula valid for fractions?
Yes. It works for fractions and decimals.
10. Can the first term be zero?
Yes. The first term can be zero.
11. Does the formula work for decreasing APs?
Yes, when d is negative, the formula still works.
12. Can the [n^{th}] term be zero?
Yes, depending on values of a, d, and n.
13. Is the [n^{th}] term always increasing?
No. It increases only when d > 0.
14. Why is [n^{th}] term important?
It helps find any term directly without writing the entire AP.
15. Is the [n^{th}] term linear in [n]?
Yes. The expression [aₙ = a + (n − 1)d] is linear in [n].
7. FAQs / Common Misconceptions
1. [n^{th}] term means the last term.
Wrong. It means the general term.
2. The formula works only for increasing APs.
Incorrect. It works for all APs.
3. [n] can be zero.
False. n ≥ 1.
4. d must be positive.
Wrong. d can be negative or zero.
5. [n^{th}] term formula gives sum of terms.
Incorrect. It gives only one term.
6. All APs have integer terms.
False. Terms can be fractional.
7. The first term must be positive.
Wrong. It can be negative or zero.
8. The formula changes for different APs.
No. The formula is universal.
9. [n^{th}] term gives position of a term.
Incorrect. It gives the value of the term.
10. [n^{th}] term is always larger than first term.
Not true. It depends on d.
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