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For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 9th and 8th terms of this sequence is 1869, what is the 11th term of this sequence?

For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 9th and 8th terms of this sequence is 1869, what is the 11th term of this sequence? Correct Answer 144

Calculation:

It is given that in a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence.

Let x and y be the 1st and 2nd term respectively.

⇒ 3rd term = x + y

⇒ 4th term = x + 2y

⇒ 5th term = 2x + 3y

⇒ 6th term = 3x + 5y

⇒ 7th term = 5x + 8y

⇒ 8th term = 8x + 13y

⇒ 9th term = 13x + 21y

We know that difference of the squares of 9th and 8th terms is 1869 = 89 × 21

And a2 - b2 = (a + b)(a - b)

Applying above formula we get (21x + 34y)(5x + 8y) = 89 × 21. So only possible way is (21x + 34y) = 89 and 5x + 8y = 21 .

Solving we get x = 1 and y = 2 .

Using the concept that every term is the sum of the previous two terms, as used at the beginning of the solution, we get the 11th term as 34x + 55y, which gives 11th term as 144.

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