Hi guys this is my first time using the forums and I need help on 1 question:
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 + ....]
I know to start the problem you somehow rewrite these infinite series but idk how. Please help I love you guys!
+23246 Using the quadratic equation to find the solutions to 7x2 -x - 3 = 0, we can get these answers:
a = [ 1 + sqrt(85) ] / 14 and b = [ 1 - sqrt(85) ] / 14
Both 1 + a + a2 + a3 + ... and 1 + b + b2 + b3 + ...
are infinite geometric series with common ratios in the range -1 < r < 1,
so we can use the formula: Sum = a1 / ( 1 - r)
In both cases, a1 = 1, so:
the sum of 1 + a + a2 + a3 + ... is 1 / [ 1 - (1 + sqrt(85))/14 ]
= 14 / [ 14 - (1 + sqrt(85) ] = 14 / [ 13 - sqrt(85) ]
and the sum of 1 + b + b2 + b3 + ... is 14 / [ 13 + sqrt(85) ]
Multiplying these two values together, we get 196 / (169 - 85) 196 / 84 = 7/3
