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please help, I am struggling with questions involving big and scary sequences.

 Mar 6, 2023
 #1
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-1

By the quadratic formula, the roots of 7x^2 - x - 3 = 0 are (1 +/- sqrt(-83))/14.

 

Then by infinite geometric series,

 

(1 + a + a^2 + ....)(1 + b + b^2 + ...) = 1/(1 - a)*1/(1 - b)\

= 1/(1 - (1 + sqrt(-83))/14))*1/(1 - (1 - sqrt(-83))/14))

= 5/2.

 Mar 6, 2023
 #2
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 To evaluate the sum of the geometric series with first term 1 and common ratio a, we can use the formula:

S = 1 + a + a^2 + a^3 + ... = 1/(1-a)

Similarly, for a geometric series with first term 1 and common ratio b, we have:

T = 1 + b + b^2 + b^3 + ... = 1/(1-b)

Using the values of a and b from the previous question, we have:

a = (1 + sqrt(85)) / 14
b = (1 - sqrt(85)) / 14


We have:

S * T = (1/(1-a))(1/(1-b))

Substituting the values of a and b, we get:

S * T = (1/(1-(1 + sqrt(85))/14)) * (1/(1-(1 - sqrt(85))/14)) ==7 / 3

 Mar 6, 2023

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