Like the Fibonacci sequence, a certain sequence satisfies the recurrence relation an=an−1+an−2. Unlike the Fibonacci sequence, however, the first two terms are a1=4 and a2=1.

Find a32.

Definition: An arithmetic sequence is a sequence of the form

a, a + d, a + 2d, a + 3d, a + 4d, …
The number a is the first term, and d is the common difference of the
sequence. The nth term of an arithmetic sequence is given by
an = a + (n – 1)d
The number d is called the common difference because any two consecutive terms of an
arithmetic sequence differ by d, and it is found by subtracting any pair of terms an and
an+1. That is
d = an+1 – an

Ah, the good old Fibonacci sequence's rebellious cousin, with the first two terms being 4 and 1. Let's see where this leads us.

To find a32, we'll need to exert our bot-brains and calculate each term along the way. I hope you're ready for some mathematical mischief!

Starting with a1 and a2, let's dive into the sequence:

a3 = a2 + a1 = 1 + 4 = 5
a4 = a3 + a2 = 5 + 1 = 6
a5 = a4 + a3 = 6 + 5 = 11
a6 = a5 + a4 = 11 + 6 = 17
a7 = a6 + a5 = 17 + 11 = 28
a8 = a7 + a6 = 28 + 17 = 45
a9 = a8 + a7 = 45 + 28 = 73

Phew, we're getting there! Keep up your patience with this sequence dance:

a10 = a9 + a8 = 73 + 45 = 118
a11 = a10 + a9 = 118 + 73 = 191
a12 = a11 + a10 = 191 + 118 = 309
a13 = a12 + a11 = 309 + 191 = 500
a14 = a13 + a12 = 500 + 309 = 809
a15 = a14 + a13 = 809 + 500 = 1309
a16 = a15 + a14 = 1309 + 809 = 2118
a17 = a16 + a15 = 2118 + 1309 = 3427
a18 = a17 + a16 = 3427 + 2118 = 5545
a19 = a18 + a17 = 5545 + 3427 = 8972
a20 = a19 + a18 = 8972 + 5545 = 14517
a21 = a20 + a19 = 14517 + 8972 = 23489
a22 = a21 + a20 = 23489 + 14517 = 38006
a23 = a22 + a21 = 38006 + 23489 = 61495
a24 = a23 + a22 = 61495 + 38006 = 99501
a25 = a24 + a23 = 99501 + 61495 = 160996
a26 = a25 + a24 = 160996 + 99501 = 260497
a27 = a26 + a25 = 260497 + 160996 = 421493
a28 = a27 + a26 = 421493 + 260497 = 682990
a29 = a28 + a27 = 682990 + 421493 = 1102483
a30 = a29 + a28 = 1102483 + 682990 = 1785473
a31 = a30 + a29 = 1785473 + 1102483 = 2887956
a32 = a31 + a30 = 2887956 + 1785473 = 4673429

And there we have it, my friend. The 32nd term of this special sequence, with naughty first two terms of 4 and 1, is a whopping 4,673,429. Bravo!

To find a32 in the given sequence, we need to apply the recurrence relation repeatedly until we reach the desired term.

Let's begin by listing the first few terms of the sequence:
a1 = 4
a2 = 1

To find a3, we use the recurrence relation:
a3 = a2 + a1 = 1 + 4 = 5

Next, we proceed to find a4:
a4 = a3 + a2 = 5 + 1 = 6

Continuing in this manner, we find the following terms of the sequence:
a5 = a4 + a3 = 6 + 5 = 11
a6 = a5 + a4 = 11 + 6 = 17
a7 = a6 + a5 = 17 + 11 = 28
a8 = a7 + a6 = 28 + 17 = 45

We can observe that the terms of this sequence are growing larger and larger. Calculating each term manually would become time-consuming.

To find a32, we can use a more efficient approach. Let's take advantage of the relationship between this sequence and the Fibonacci sequence.

If we look closely, we realize that this sequence is simply a Fibonacci-like sequence with different initial conditions. The Fibonacci sequence starts with a0 = 0, a1 = 1, and uses the same recurrence relation as the given sequence: an = an-1 + an-2.

To relate our sequence to the Fibonacci sequence, we need to shift the indices by 1. If we denote the Fibonacci sequence as F0, F1, F2, F3, ..., then the corresponding terms of our sequence can be denoted as a1, a2, a3, a4, ... respectively.

Therefore, a32 would correspond to the 34th term in the Fibonacci sequence (F33), since a1 corresponds to F1, a2 corresponds to F2, and so on.

Using the well-known formula for the nth term of the Fibonacci sequence:
F(n) = [(phi^n) - ((-phi)^(-n))] / sqrt(5)

(where phi is the golden ratio, approximately 1.6180339887)

We can calculate F33:

F(33) = [(phi^33) - ((-phi)^(-33))] / sqrt(5)

Now, let's plug in the values and calculate it using a calculator or computer program:

F(33) ≈ 3,524,578

Therefore, a32 in the given sequence is approximately 3,524,578.

Heck, just start listing the terms:

4,1,5,6,11,17,28,...

It's not a big job to find the 32nd term.

I don't know