The sequence {an} satisfies a0=1,a1=213, and an=2an−1+an−2 for all n≥2. Let
S=∑i=1 to infinity a[i-1]/(a[i]^2-a[i-1]^2)
What is the value of 1/S?
148
148 is incorrect
424
To find the value of 1/S, we first need to determine the values of the terms in the sequence {an}.
Given that a0 = 1 and a1 = 213, we can use the recurrence relation an = 2an−1 + an−2 for n ≥ 2 to find the subsequent terms of the sequence.
Let's calculate the first few terms:
a2 = 2a1 + a0 = 2(213) + 1 = 427 + 1 = 428
a3 = 2a2 + a1 = 2(428) + 213 = 856 + 213 = 1069
a4 = 2a3 + a2 = 2(1069) + 428 = 2138 + 428 = 2566
...
We can continue this process to find more terms if necessary.
Next, we need to find the value of S by evaluating the infinite series:
S = ∑i=1 to infinity a[i-1]/(a[i]^2-a[i-1]^2)
Since the series is infinite, we cannot directly calculate its value. However, we can use some properties of geometric series to simplify the expression.
Note that the denominator of each term can be factored as a difference of squares:
a[i]^2 - a[i-1]^2 = (a[i] - a[i-1])(a[i] + a[i-1])
Let's substitute this into the expression for S:
S = ∑i=1 to infinity a[i-1]/((a[i] - a[i-1])(a[i] + a[i-1]))
Next, we can cancel out the common factors in the numerator and denominator of each term:
S = ∑i=1 to infinity 1/(a[i] + a[i-1])
Now, let's simplify this further by grouping the terms:
S = (1/(a1 + a0)) + (1/(a2 + a1)) + (1/(a3 + a2)) + ...
Notice that each term is a ratio of consecutive terms in the sequence {an}. Since we've already calculated the terms of the sequence, we can substitute the values:
S = (1/(213 + 1)) + (1/(428 + 213)) + (1/(1069 + 428)) + ...
Now, we can calculate the individual terms and add them up:
S = 1/214 + 1/641 + 1/1497 + ...
To find the value of 1/S, we need to invert the sum:
1/S = 1/(1/214 + 1/641 + 1/1497 + ...)
To simplify this, we need to find a common denominator for the fractions in the sum. The common denominator will be the least common multiple (LCM) of the denominators.
The LCM of 214, 641, 1497, ... can be calculated using prime factorization or a calculator. Once we have the LCM, we can rewrite each fraction with the common denominator and then add them up.
Finally, we can take the reciprocal of the sum to find the value of 1/S.
Unfortunately, without knowing the terms of the sequence beyond what we've calculated, we cannot determine the exact value of 1/S or the sequence. The problem requires more information or a specific pattern for the terms of the sequence to find a precise solution.