A series in which any term is the sum of the preceding two terms is ca...
The series is like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
The difference is 715 and the 12th term is 144.
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A series in which any term is the sum of the preceding two terms is ca...
Given:
- The difference of the squares of the ninth and the eighth terms of a Fibonacci series is 715.
To find:
- The 12th term of that series.
Approach:
- We can use the formula to find the nth term of a Fibonacci series: Fn = Fn-1 + Fn-2
- We can also use the given information to form equations and solve for the required term.
Solution:
Let's assume that the first two terms of the Fibonacci series are a and b, then we can write the series as:
a, b, a+b, 2b+a, 3b+2a, 5b+3a, 8b+5a, 13b+8a, 21b+13a, ...
We are given that the difference of the squares of the ninth and the eighth terms of the series is 715, so we can write the equation as follows:
(21b+13a)^2 - (13b+8a)^2 = 715
Simplifying the equation, we get:
(8a+21b+13a+13b)(8a+21b-13a-13b) = 715
(21a+34b)(-5a+8b) = 715
We can find two factors of 715 whose difference is 13*2*5 = 130, which are 65 and 11:
(21a+34b) = 65 and (-5a+8b) = 11
Solving these equations, we get: a = 2 and b = 1.
Now, we can use the formula to find the 12th term of the series:
F12 = F11 + F10
F10 = F9 + F8
F11 = F10 + F9
Using these equations, we can find that F9 = 34, F8 = 21, F10 = 55, and F11 = 89.
Therefore, F12 = F11 + F10 = 89 + 55 = 144.
Hence, the correct answer is option (c) 144.