Finite geometric sequence

Find the remaining unknown characteristics in the finite geometric sequence, if given: a1 = 18, an = 13122, sn = 19674, n =? q =?

Correct answer:

n = 7
q = 3

Step-by-step explanation:

a_{1} = 18 a = 13122 s = 19674 a = a_{1} \cdot q^{n - 1} s = a_{1} \cdot \frac{q ^{n} - 1}{q - 1} a \cdot q = a_{1} \cdot q^{n} a \cdot q - a_{1} = a_{1} \cdot (q^{n} - 1) s = a_{1} \cdot \frac{q ^{n} - 1}{q - 1} = a_{1} \cdot \frac{( a \cdot q - a _{1} ) / a _{1}}{q - 1} s = \frac{a \cdot q - a _{1}}{q - 1} s \cdot (q - 1) = a \cdot q - a_{1} 19674 \cdot (q - 1) = 13122 \cdot q - 18 6552 q = 19656 q = \frac{1 9 6 5 6}{6 5 5 2} = 3 q = 3 a \cdot q = a_{1} \cdot q^{n} ln (a \cdot q / a_{1}) = n \cdot ln q n = \frac{ln ( a \cdot q / a _{1} )}{ln q} = \frac{ln ( 1 3 1 2 2 \cdot 3 / 1 8 )}{ln 3} = 7

q = 3 Verifying Solution: a_{2} = q \cdot a_{1} = 3 \cdot 18 = 54 a_{3} = q \cdot a_{2} = 3 \cdot 54 = 162 a_{4} = q \cdot a_{3} = 3 \cdot 162 = 486 a_{5} = q \cdot a_{4} = 3 \cdot 486 = 1458 a_{6} = q \cdot a_{5} = 3 \cdot 1458 = 4374 a_{7} = q \cdot a_{6} = 3 \cdot 4374 = 13122 s_{7} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} = 18 + 54 + 162 + 486 + 1458 + 4374 + 13122 = 19674

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Grade of the word problem:

high school

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