# Geometric sequence

In the geometric sequence is a4 = 20 a9 = -160. Calculate the first member a1 and quotient q.

Result

q =  -1.516
a1 =  -5.743

#### Solution:

$a_{ 9 } = a_1 q^{ 8} = -160 \ \\ a_{ 4 } = a_1 q^{ 3} = 20 \ \\ \ \\ \dfrac{ -160 }{ 20 } = \dfrac{ q^{ 8} }{ q^{ 3} } \ \\ \ \\ -8 = q^{ 5} \ \\ (-8)^{ 0.2 } = q \ \\ q = -1.516$
$a_{ 9 } = a_1 q^{ 8} \ \\ a_1 = \dfrac{ -160 }{ q ^{ 8}} = -5.743 \ \\ a_{ 2 } = a_{ 1 } \cdot q = 8.7055056329612 \ \\ a_{ 3 } = a_{ 2 } \cdot q = -13.195079107729 \ \\ a_{ 4 } = a_{ 3 } \cdot q = 20 \ \\ a_{ 5 } = a_{ 4 } \cdot q = -30.314331330208 \ \\ a_{ 6 } = a_{ 5 } \cdot q = 45.947934199881 \ \\ a_{ 7 } = a_{ 6 } \cdot q = -69.64404506369 \ \\ a_{ 8 } = a_{ 7 } \cdot q = 105.56063286183 \ \\ a_{ 9 } = a_{ 8 } \cdot q = -160 \ \\ a_{ 10 } = a_{ 9 } \cdot q = 242.51465064166 \ \\ a_{ 11 } = a_{ 10 } \cdot q = -367.58347359905 \ \\ a_{ 12 } = a_{ 11 } \cdot q = 557.15236050952 \ \\ a_{ 13 } = a_{ 12 } \cdot q = -844.48506289465 \ \\ a_{ 14 } = a_{ 13 } \cdot q = 1280 \ \\ a_{ 15 } = a_{ 14 } \cdot q = -1940.1172051333 \ \\ a_{ 16 } = a_{ 15 } \cdot q = 2940.6677887924 \ \\ a_{ 17 } = a_{ 16 } \cdot q = -4457.2188840762 \ \\ a_{ 18 } = a_{ 17 } \cdot q = 6755.8805031572 \ \\ a_{ 19 } = a_{ 18 } \cdot q = -10240 \ \\ a_{ 20 } = a_{ 19 } \cdot q = 15520.937641066 \ \\ a_{ 21 } = a_{ 20 } \cdot q = -23525.342310339 \ \\ a_{ 22 } = a_{ 21 } \cdot q = 35657.751072609$

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