The Fibonacci sequence is a famous sequence in mathematics where each number is the sum of the two preceding ones, usually starting with 1 and 1. We denote the sequence by \( f(n) \), where \( f(n) \) represents the \( n \)-th term in the sequence. The general recurrence relation for the Fibonacci sequence is given by:
[tex]\[ f(n+1) = f(n) + f(n-1) \][/tex]
The initial conditions are \( f(1) = 1 \) and \( f(2) = 1 \).
Given this, let's see how each term in the Fibonacci sequence up to the 10th term is generated:
1. First Term ( \( f(1) \) ):
[tex]\[
f(1) = 1
\][/tex]
2. Second Term ( \( f(2) \) ):
[tex]\[
f(2) = 1
\][/tex]
3. Third Term ( \( f(3) \) ):
Applying the recurrence relation:
[tex]\[
f(3) = f(2) + f(1) = 1 + 1 = 2
\][/tex]
4. Fourth Term ( \( f(4) \) ):
[tex]\[
f(4) = f(3) + f(2) = 2 + 1 = 3
\][/tex]
5. Fifth Term ( \( f(5) \) ):
[tex]\[
f(5) = f(4) + f(3) = 3 + 2 = 5
\][/tex]
6. Sixth Term ( \( f(6) \) ):
[tex]\[
f(6) = f(5) + f(4) = 5 + 3 = 8
\][/tex]
7. Seventh Term ( \( f(7) \) ):
[tex]\[
f(7) = f(6) + f(5) = 8 + 5 = 13
\][/tex]
8. Eighth Term ( \( f(8) \) ):
[tex]\[
f(8) = f(7) + f(6) = 13 + 8 = 21
\][/tex]
9. Ninth Term ( \( f(9) \) ):
[tex]\[
f(9) = f(8) + f(7) = 21 + 13 = 34
\][/tex]
10. Tenth Term ( \( f(10) \) ):
[tex]\[
f(10) = f(9) + f(8) = 34 + 21 = 55
\][/tex]
Therefore, the first ten terms of the Fibonacci sequence are:
[tex]\[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 \][/tex]
Each term is derived by summing the two preceding terms, starting from the initial values 1 and 1.
