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Find the 10^("th ") term of the following geometric sequence. 3,15,75,375,dots

Find the 10th  10^{\text {th }} term of the following geometric sequence.\newline3,15,75,375,… 3,15,75,375, \ldots

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Full solution

Q. Find the 10th  10^{\text {th }} term of the following geometric sequence.\newline3,15,75,375,… 3,15,75,375, \ldots
  1. Find Common Ratio: Identify the common ratio rr of the geometric sequence.\newlineTo find the common ratio, we divide the second term by the first term.\newliner=153=5r = \frac{15}{3} = 5
  2. 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:\newlinean=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}\newlinewhere a1a_1 is the first term and rr is the common ratio.
  3. Calculate 1010th Term: Calculate the 1010th term using the formula.\newlinea10=a1×r10−1a_{10} = a_1 \times r^{10-1}\newlinea10=3×510−1a_{10} = 3 \times 5^{10-1}\newlinea10=3×59a_{10} = 3 \times 5^9
  4. Perform Exponentiation: Perform the exponentiation.\newline59=5×5×5×5×5×5×5×5×55^9 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\newline59=19531255^9 = 1953125
  5. Multiply First Term: Multiply the first term by the result of the exponentiation.\newlinea10=3×1953125a_{10} = 3 \times 1953125\newlinea10=5859375a_{10} = 5859375

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