Q. Find the 13th term of the geometric sequence 3,−9,27,…Answer:
Identify First Term: To find the 13th term of a geometric sequence, we need to use the formula for the nth term of a geometric sequence, which is an=a1⋅r(n−1), where a1 is the first term, r is the common ratio, and n is the term number.
Find Common Ratio: First, we identify the first term a1 of the sequence, which is 3.
Calculate 13th Term Formula: Next, we need to find the common ratio r. We can do this by dividing the second term by the first term: r=3−9=−3.
Calculate Exponent: Now that we have the first term and the common ratio, we can find the 13th term a13 using the formula: a13=3×(−3)13−1.
Simplify Exponent Calculation: We calculate the exponent: (−3)(13−1)=(−3)12. Since (−3)12 is a positive number because 12 is an even exponent, we can proceed with the calculation.
Calculate Large Exponent: Now we calculate (−3)12. This is a large number, but we can simplify the calculation by breaking it down: (−3)12=((−3)2)6=96.
Calculate Final Term: Next, we calculate 96. This can be done by multiplying 9 by itself 6 times: 96=9×9×9×9×9×9=531441.
Calculate Final Term: Next, we calculate 96. This can be done by multiplying 9 by itself 6 times: 96=9×9×9×9×9×9=531441. Finally, we multiply the first term by the result of the exponent calculation to find the 13th term: a13=3×531441=1594323.
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