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Find the 9^th term of the geometric sequence 2,6,18,dots Answer:

Find the 9th 9^{\text {th }} term of the geometric sequence 2,6,18,… 2,6,18, \ldots \newlineAnswer:

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Full solution

Q. Find the 9th 9^{\text {th }} term of the geometric sequence 2,6,18,… 2,6,18, \ldots \newlineAnswer:
  1. Identify Sequence Type: Identify the type of sequence. The given sequence is geometric because each term after the first is found by multiplying the previous term by a constant called the common ratio.
  2. Determine Common Ratio: Determine the common ratio rr of the sequence.\newlineTo find the common ratio, divide the second term by the first term.\newliner=62=3r = \frac{6}{2} = 3
  3. Use Formula for nth Term: Use the formula for the nth term of a geometric sequence.\newlineThe nth term ana_n of a geometric sequence can be found using the formula an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term and rr is the common ratio.
  4. Calculate 99th Term: Calculate the 99th term using the formula.\newlinea9=a1⋅r9−1a_9 = a_1 \cdot r^{9-1}\newlinea9=2⋅39−1a_9 = 2 \cdot 3^{9-1}\newlinea9=2⋅38a_9 = 2 \cdot 3^8
  5. Perform Exponentiation: Perform the exponentiation. 38=3×3×3×3×3×3×3×3=65613^8 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561
  6. Multiply First Term: Multiply the first term by the result of the exponentiation to get the 9th9^{\text{th}} term.\newlinea9=2×6561a_9 = 2 \times 6561\newlinea9=13122a_9 = 13122

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