Step (1)
We know that the sum of all the terms in an infinite geometric sequence finding from following equation whenever |q| < 1 .
q = magnitude
[tex]s( \infty ) = \frac{t(1)}{1 - q} \\ [/tex]
So :
[tex]s( \infty ) = \frac{t(1)}{1 - q} \\ [/tex]
[tex]81 = \frac{t(1)}{1 - q} \\ [/tex]
[tex]81(1 - q) = t(1)[/tex]
Remember it I'll use it again.
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Step (2)
We know that the sum of the n first terms of a geometric sequence finding from following equation.
[tex]s(n) = \frac{t(1) \times (1 - {q})^{n} }{1 - q} \\ [/tex]
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Sum of all terms starting from the third is 9.
So :
[tex]s(2) + s(3 - \infty ) = 81[/tex]
[tex]s(2) + 9 = 81[/tex]
Sides minus 9
[tex]s(2) = 81 - 9[/tex]
[tex]s(2) = 72[/tex]
So :
[tex]t(1) + t(2) = 72[/tex]
[tex]t(1) + t(1) \times q = 72[/tex]
Factoring t(1)
[tex]t(1) \times (1 + q) = 72[/tex]
We have found t(1) = 81 ( 1 - q ) in step (1).
So :
[tex]81(1 - q)(1 + q) = 72[/tex]
Divided sides by 81
[tex](1 - q)(1 + q) = \frac{72}{81} \\ [/tex]
[tex](1 - q)( 1 + q) = \frac{8}{9} \\ [/tex]
[tex]1 - {q}^{2} = \frac{8}{9} \\ [/tex]
Subtract sides minus -1
[tex] - {q}^{2} = \frac{8}{9} - 1 \\ [/tex]
[tex] - {q}^{2} = \frac{8}{9} - \frac{9}{9} \\ [/tex]
[tex] - {q}^{2} = - \frac{1}{9} \\ [/tex]
Negatives simplifies
[tex] {q}^{2} = \frac{1}{9} \\ [/tex]
Radical sides
[tex]q = + \frac{1}{3} \\ q = - \frac{1}{3} [/tex]
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Step (4)
If q = 1/3 :
[tex]81(1 - q) = t(1)[/tex]
[tex]81(1 - \frac{1}{3}) = t(1) \\ [/tex]
[tex]81( \frac{2}{3}) = t(1) \\ [/tex]
[tex]t(1) = 27 \times 2[/tex]
[tex]t(1) = 54[/tex]
So :
[tex]t(2) = t(1) \times q[/tex]
[tex]t(2) = 54 \times \frac{1}{3} \\ [/tex]
[tex]t(2) = 18[/tex]
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
If q = - 1/3 :
[tex]81(1 - q) = t(1)[/tex]
[tex]81(1 - ( - \frac{1}{3})) = t(1) \\ [/tex]
[tex]81(1 + \frac{1}{3}) = t(1) \\ [/tex]
[tex]81( \frac{4}{3}) = t(1) \\ [/tex]
[tex]t(1) = 27 \times 4[/tex]
[tex]t(1) = 108[/tex]
So :
[tex]t(2) = t(1) \times q[/tex]
[tex]t(2) = 108 \times - \frac{1}{3} \\ [/tex]
[tex]t(2) = - 36[/tex]
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And we're done.
Thanks for watching buddy good luck.
♥️♥️♥️♥️♥️
