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Nth Term — Definition, Formula & Examples

The nth term is a rule that gives the value of any term in a sequence from its position. You plug in a position number nn and get the term in that spot.

For a sequence (an)(a_n), the nth term (or general term) ana_n is the value the sequence assigns to the index nn, where nn is a positive integer (sometimes starting at n=0n = 0). An explicit formula an=f(n)a_n = f(n) defines the sequence as a function whose domain is the set of positive integers, for example an=2n−1a_n = 2n - 1 or an=1na_n = \frac{1}{n}.

Worked Example

Problem: Find the nth term of the sequence 4, 7, 10, 13, ... Then use it to find the 50th term.
Step 1: Subtract consecutive terms to check the pattern. Each difference is the same, so the sequence is arithmetic with first term a1=4a_1 = 4 and common difference d=3d = 3.
d=7−4=3d = 7 - 4 = 3
Step 2: Substitute into the arithmetic rule an=a1+(n−1)da_n = a_1 + (n-1)d and simplify. Check it with n=1n = 1: 3(1)+1=43(1) + 1 = 4, which matches the first term.
an=4+(n−1)(3)=3n+1a_n = 4 + (n-1)(3) = 3n + 1
Step 3: Substitute n=50n = 50 to find the 50th term without listing all 50 terms.
a50=3(50)+1=151a_{50} = 3(50) + 1 = 151
Answer: The nth term is an=3n+1a_n = 3n + 1, and the 50th term is a50=151a_{50} = 151.

Why It Matters

In precalculus and calculus you study what ana_n does as n→∞n \to \infty. That behavior decides whether a sequence converges or diverges. In Calculus II, the nth term test for divergence says a series cannot converge unless its nth term approaches 0.

Common Mistakes

Mistake: Writing the rule as an=a1+nda_n = a_1 + nd, for example an=4+3na_n = 4 + 3n for the sequence 4, 7, 10, ...
Correction: That rule gives a1=7a_1 = 7, which is the second term. The first term has had zero differences added, so use (n−1)d(n-1)d. Always test your rule at n=1n = 1.

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