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Find the 13^("th ") term of the geometric sequence 6,18,54,dots Answer:

Find the 13th  13^{\text {th }} term of the geometric sequence 6,18,54,… 6,18,54, \ldots \newlineAnswer:

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Full solution

Q. Find the 13th  13^{\text {th }} term of the geometric sequence 6,18,54,… 6,18,54, \ldots \newlineAnswer:
  1. Identify first term: To find the 13th13^{\text{th}} term of a geometric sequence, we need to use the formula for the nthn^{\text{th}} term of a geometric sequence, which is an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term, rr is the common ratio, and nn is the term number.
  2. Find common ratio: First, we identify the first term a1a_1 of the sequence, which is 66.
  3. Calculate 1313th term: Next, we need to find the common ratio rr. We can do this by dividing the second term by the first term, or the third term by the second term. Let's use the second term divided by the first term: r=186=3r = \frac{18}{6} = 3.
  4. Calculate power of ratio: Now that we have the first term and the common ratio, we can find the 1313th term a13a_{13} using the formula: a13=a1×r(13−1)=6×312a_{13} = a_1 \times r^{(13-1)} = 6 \times 3^{12}.
  5. Multiply to find term: We calculate 3123^{12}. 312=3×3×3×3×3×3×3×3×3×3×3×3=531,4413^{12} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 531,441.
  6. Multiply to find term: We calculate 3123^{12}. 312=3×3×3×3×3×3×3×3×3×3×3×3=531,4413^{12} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 531,441.Finally, we multiply the first term by 3123^{12} to find the 1313th term: a13=6×531,441=3,188,646a_{13} = 6 \times 531,441 = 3,188,646.

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