in the number sequence 1,1,2,3,5,8,13.... What is the remainder when the 1995th number is divided by 8?
+33661 Another way to see this is to note that the remainders of the Fibonacci sequence divided by 8 follow a pattern that repeats every 12'th number - see below. Hence, to find the remainder of the 1995'th term, find the remainder of 1995 divided by 12, then use this as the index into the 12 remaindwers of the Fibonacci sequence.
Since mod(1995,12) = 3, look at the third of the 12 repeating values. As can be seen from the image below, this is 2.
.
+333 Use binet's formula to find the 1995th term of the series hint: it's the fibonnacia series
so \(F(n)=\frac{\phi^n-(-\phi)^n}{\sqrt5}\)Let n=1995
\(F(1995)=\frac{\phi^{1995}-(-\phi)^{1995}}{\sqrt{5}}\)
Let Phi=1.6 \(F(1995)=23668326.9616766279790584\) divided by 8 it equals
\(=2958540.8702095784973823\)
+26393 in the number sequence 1,1,2,3,5,8,13.... What is the remainder when the 1995th number is divided by 8?
\(f_{1995} =\)
\(f_{1995} \pmod 8 = 2\)
+33661 Another way to see this is to note that the remainders of the Fibonacci sequence divided by 8 follow a pattern that repeats every 12'th number - see below. Hence, to find the remainder of the 1995'th term, find the remainder of 1995 divided by 12, then use this as the index into the 12 remaindwers of the Fibonacci sequence.
Since mod(1995,12) = 3, look at the third of the 12 repeating values. As can be seen from the image below, this is 2.
.
