Question

A series in which any term is equal to the sum of the preceding two terms is called a Fibonacci series. Usually the first two terms are given initially and together they determine the entire series. Now, it is known that the difference of the squares of the ninth and the eighth terms of a Fibonacci series is 840. What is the ${12}^{th}$ term of that series?

• A. 157
• B. 142
• C. 143
• D. Cannot be determined

Answer 1

The correct option is D Cannot be determined

$Leta,b,a+b,a+2b,2a+3b....$ be the Fibonacci series.

$(9^{th}term{)}^2-(8^{th}term{)}^2=840(13a+21b{)}^2-(8a+13b{)}^2=840105a^2+338ab+272b^2=840(5a+8b)(21a+34b)=84012^{th}term=55a+89bIfa=1,b=1(5a+8b)(21a+34b)=715Ifa=2,b=1\left(or\right)a=1,b=2(5a+8b)(21a+34b)>840$

It is not possible and ${12}^{th}$term cannot be determined.