GP sequence

Find the remaining unknown characteristics in the finite geometric sequence, if given: a1 = 5, an = 320, sn = 635, n =? q =?

Final Answer:

n = 7
q = 2

a_{1} = 5 a = 320 s = 635 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} 635 \cdot (q - 1) = 320 \cdot q - 5 315 q = 630 q = \frac{6 3 0}{3 1 5} = 2 q = 2 a \cdot q = a_{1} \cdot q^{n} a \cdot q / a_{1} = q^{n - 1} 64 = 2^{n - 1} 2^{6} = 2^{n - 1} n = 6 + 1 = 7 n = \frac{ln ( a \cdot q / a _{1} )}{ln q}

q = 2 Verifying Solution: a_{2} = q \cdot a_{1} = 2 \cdot 5 = 10 a_{3} = q \cdot a_{2} = 2 \cdot 10 = 20 a_{4} = q \cdot a_{3} = 2 \cdot 20 = 40 a_{5} = q \cdot a_{4} = 2 \cdot 40 = 80 a_{6} = q \cdot a_{5} = 2 \cdot 80 = 160 a_{7} = q \cdot a_{6} = 2 \cdot 160 = 320 s_{7} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} = 5 + 10 + 20 + 40 + 80 + 160 + 320 = 635

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