Finding the n-th Term of an Arithmetic Sequence
Find any term of an arithmetic sequence in Grade 9 algebra using the explicit formula aₙ=a₁+(n-1)d, where a₁ is the first term and d is the common difference between consecutive terms.
Key Concepts
Property Use the explicit formula $a n = a 1 + (n 1)d$ to find any term ($a n$) in an arithmetic sequence, using the first term ($a 1$) and the common difference ($d$).
Examples To find the $11^{\text{th}}$ term where $a 1=6$ and $d=2$: $a {11} = 6 + (11 1)2 = 6 + (10)2 = 26$. To find the $10^{\text{th}}$ term of $3, 11, 19, \dots$: First, find $d = 11 3=8$. Then $a {10} = 3 + (10 1)8 = 75$. To find the $10^{\text{th}}$ term of $\frac{1}{4}, \frac{3}{4}, \dots$: First, find $d=\frac{1}{2}$. Then $a {10} = \frac{1}{4} + (10 1)\frac{1}{2} = \frac{19}{4}$.
Explanation This formula is a fantastic shortcut. It lets you leap directly to any term in the sequence, like the 100th, without finding the 99 terms before it. Just plug in your starting point, your common difference, and which term number you want to find. It’s a direct flight to your answer!
Common Questions
What is the explicit formula for the nth term of an arithmetic sequence?
The explicit formula is 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. This formula gives any term directly without listing all previous terms.
How do you find the 20th term of the sequence 5, 8, 11, 14, ...?
Identify a₁ = 5 and d = 3 (difference between consecutive terms). Apply the formula: a₂₀ = 5 + (20-1)(3) = 5 + 57 = 62. The 20th term is 62.
How do you find which term in a sequence equals a specific value?
Set the explicit formula equal to the target value and solve for n. For the sequence aₙ = 5 + (n-1)(3), find n when aₙ = 44: 5 + 3(n-1) = 44 → 3(n-1) = 39 → n-1 = 13 → n = 14. The 14th term equals 44.