Find the Nth Term Using Explicit Formula

Simple

• Question: A geometric sequence has a first term $a_1 = 8$ and a common ratio $r = -\frac{1}{2}$. Find the 7th term.
• Solution:
  • Given $a_1 = 8$, $r = -\frac{1}{2}$, $n = 7$.
  • Use the formula: $a_7 = 8 \cdot \left(-\frac{1}{2}\right)^{7-1} = 8 \cdot \left(-\frac{1}{2}\right)^6$.
  • Simplify: $\left(-\frac{1}{2}\right)^6 = \frac{1}{64}$, so $a_7 = 8 \cdot \frac{1}{64} = \frac{1}{8}$.
• Conclusion: The 7th term is $\frac{1}{8}$.

Medium

• Question: Find the 6th term of the geometric sequence $2, 6, 18, 54, ...$ using the explicit formula.
• Solution:
  • Identify $a_1 = 2$, $r = \frac{6}{2} = 3$, $n = 6$.
  • Apply the formula: $a_6 = a_1 \cdot r^{6-1} = 2 \cdot 3^5$.
  • Calculate: $3^5 = 243$, so $a_6 = 2 \cdot 243 = 486$.
• Conclusion: The 6th term is 486.

Hard

• Question: A geometric sequence has a 3rd term $a_3 = 18$ and a common ratio $r = 3$. Find the 10th term.
• Solution:
  • Step 1: Find $a_1$ using the 3rd term. From $a_3 = a_1 \cdot r^{3-1}$, substitute $a_3 = 18$ and $r = 3$:$18 = a_1 \cdot 3^2 \implies 18 = a_1 \cdot 9 \implies a_1 = 2$.
  • Step 2: Calculate the 10th term with $a_1 = 2$, $r = 3$, $n = 10$:$a_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9$.
  • Simplify: $3^9 = 19683$, so $a_{10} = 2 \cdot 19683 = 39366$.
• Conclusion: The 10th term is 39366.