Let a and b be the roots of 7x^2 - x - 3 = 0. Find (1 + a + a^2 + a^3 +.....) (1 + b + b^2 + b^3 +.....).

Thx :)

Guest Feb 26, 2022

#1**+1 **

Hint:

1) Use the quadratic formula to find the two solutions to the equation: 7x^{2} - x - 3 = 0.

Leave them in reduced radical form.

Label one solution as 'a' and the other as 'b'.

Note that each solution is in the range: -1 < x < 1,

2) 1 + a + a^{2} + a^{3} + ... is an infinite geometric series, whose sum can be found

using the formula: Sum = (first term) / [ 1 - (common ratio) ]

the first term is 1, and the common ratio is the solution that you found in step 1.

This formula applies because the common ration is in the range -1 < r < 1.

3) You can find the sum of 1 + + b^{2} + b^{3} + ... in a similar way and multiply these

two answers.

geno3141 Feb 26, 2022