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Find the 
13^("th ") term of the geometric sequence 
3,-9,27,dots
Answer:

Find the 13th  13^{\text {th }} term of the geometric sequence 3,9,27, 3,-9,27, \ldots \newlineAnswer:

Full solution

Q. Find the 13th  13^{\text {th }} term of the geometric sequence 3,9,27, 3,-9,27, \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=a1r(n1)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 33.
  3. Calculate 1313th Term Formula: Next, we need to find the common ratio rr. We can do this by dividing the second term by the first term: r=93=3r = \frac{-9}{3} = -3.
  4. Calculate Exponent: Now that we have the first term and the common ratio, we can find the 1313th term a13a_{13} using the formula: a13=3×(3)131a_{13} = 3 \times (-3)^{13-1}.
  5. Simplify Exponent Calculation: We calculate the exponent: (3)(131)=(3)12(-3)^{(13-1)} = (-3)^{12}. Since (3)12(-3)^{12} is a positive number because 1212 is an even exponent, we can proceed with the calculation.
  6. Calculate Large Exponent: Now we calculate (3)12(-3)^{12}. This is a large number, but we can simplify the calculation by breaking it down: (3)12=((3)2)6=96(-3)^{12} = ((-3)^2)^6 = 9^6.
  7. Calculate Final Term: Next, we calculate 969^6. This can be done by multiplying 99 by itself 66 times: 96=9×9×9×9×9×9=5314419^6 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 531441.
  8. Calculate Final Term: Next, we calculate 969^6. This can be done by multiplying 99 by itself 66 times: 96=9×9×9×9×9×9=5314419^6 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 531441. Finally, we multiply the first term by the result of the exponent calculation to find the 1313th term: a13=3×531441=1594323a_{13} = 3 \times 531441 = 1594323.

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