Find the first four terms of each sequence.
A1= 2
A2=5
An=An-1 + An-2, n≥3
This is a recursive relationship very similar to the Fibonacci numbers, except that A1=2 and A2=5, whereas F1=1 and F2=1 for Fibonacci numbers.
We start with
An=An-1+An-2
Substitute An-1=An-2+An-3, we get
An=2An-2+An-3
Similarly, and continuing,
An=3An-3+2An-4
An=5An-4+3An-5
An=8An-5+5An-6
An=11An-6+8An-5
....
Replacing the coefficients by Fibonacci numbers, where
F0=0, F1=1, F2=1, F3=2, F4=3...
and continuing the substitution until A2 and A1, we would get:
An=Fn-2*A1+Fn-1*A2
Thus
A4=F2*A1+F3*A2
=1*2+2*5
=12
A8
=F6*2+F7*5
=8*2+13*5
=16+65
=81
Work out A8 manually as a check.
Sure, let's find the first four terms of the sequence A1 = 2, A2 = 5, and An = An-1 + An-2, for n ≥ 3.
A1 = 2
A2 = 5
To find A3, we use the formula An = An-1 + An-2:
A3 = A2 + A1 = 5 + 2 = 7
To find A4, we once again use the formula:
A4 = A3 + A2 = 7 + 5 = 12
So, the first four terms of the sequence are:
A1 = 2, A2 = 5, A3 = 7, and A4 = 12.
To find the first four terms of the given sequence, we will use the recursive formula provided.
First, let's list the given terms A1 and A2:
A1 = 2
A2 = 5
To find A3, we can use the given recursive formula:
A3 = A3-1 + A3-2 = A2 + A1 = 5 + 2 = 7
Similarly, we can find A4:
A4 = A4-1 + A4-2 = A3 + A2 = 7 + 5 = 12
Therefore, the first four terms of the sequence are:
A1 = 2
A2 = 5
A3 = 7
A4 = 12
To find the first four terms of the sequence defined by the formula An = An-1 + An-2, where A1 = 2 and A2 = 5, we need to apply the recursive formula.
Step 1: Determine the values of A3 and A4 using the given formula.
A3 = A3-1 + A3-2 = A2 + A1 = 5 + 2 = 7
A4 = A4-1 + A4-2 = A3 + A2 = 7 + 5 = 12
Step 2: Write down the values of A1, A2, A3, and A4.
A1 = 2
A2 = 5
A3 = 7
A4 = 12
Therefore, the first four terms of the sequence are 2, 5, 7, and 12.