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Find the 13^("th ") term of the geometric sequence 3,-12,48,dots Answer:

Find the 13th  13^{\text {th }} term of the geometric sequence 3,−12,48,… 3,-12,48, \ldots \newlineAnswer:

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Q. Find the 13th  13^{\text {th }} term of the geometric sequence 3,−12,48,… 3,-12,48, \ldots \newlineAnswer:
  1. Identify type and first term: Identify the type of sequence and the first term (a1a_1).\newlineThe sequence 3,−12,48,extellipsis3, -12, 48, extellipsis is a geometric sequence because each term is multiplied by a common ratio to get the next term. The first term is a1=3a_1 = 3.
  2. Find common ratio: Find the common ratio rr of the sequence.\newlineTo find the common ratio, divide the second term by the first term: r=(−12)/3=−4r = (-12) / 3 = -4.
  3. Use nth term formula: Use the formula for the nth term of a geometric sequence.\newlineThe formula for the nth term ana_n of a geometric sequence is an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n - 1)}.
  4. Substitute values for 1313th term: Substitute the values of a1a_1 and rr into the formula to find the 1313th term.\newlinea13=3×(−4)13−1=3×(−4)12.a_{13} = 3 \times (-4)^{13 - 1} = 3 \times (-4)^{12}.
  5. Calculate 1313th term: Calculate the 13th13^{\text{th}} term.\newlineSince (−4)12(-4)^{12} is a positive number because 1212 is an even exponent, we have a13=3×(412)a_{13} = 3 \times (4^{12}).
  6. Compute 4124^{12}: Compute 4124^{12}.412=42×42×42×42×42×42=16×16×16×16×16×16=256×256×256=65536×256=167772164^{12} = 4^2 \times 4^2 \times 4^2 \times 4^2 \times 4^2 \times 4^2 = 16 \times 16 \times 16 \times 16 \times 16 \times 16 = 256 \times 256 \times 256 = 65536 \times 256 = 16777216.
  7. Multiply to get 1313th term: Multiply 33 by 1677721616777216 to get the 1313th term.\newlinea13=3×16777216=50331648a_{13} = 3 \times 16777216 = 50331648.

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