Suppose , and for , . Find the values of . Do you recognise this sequence?

Understand the Question We are given the first two terms: and . For any , the term is given by the formula: We compute each term sequentially from to and examine the resulting pattern. Step 1 · Compute terms through Given and . For : For : For : For : For : For : Step 2 · Identify the Pattern The generated sequence of numbers is: Notice that subtracting consecutive terms gives: Each term after the second is the sum of the preceding two terms. This is the Fibonacci sequence (starting from ). Answer > ; this is the Fibonacci sequence. Common Mistakes Index Off-by-One: Adding terms up to instead of . The formula specifically sums terms up to index before adding . Forgetting the constant: Omitting the final constant when summing the preceding terms.

Predicting What Comes Next: Sequences and Progressions | EOT

Question 15

Suppose $W_1 = 1$ , $W_2 = 2$ and for $n > 2$ , $W_n = W_1 + W_2 + \dots + W_{n-2} + 2$ . Find the values of $W_1, W_2, \dots, W_8$ . Do you recognise this sequence?

Solution

Understand the Question

We are given the first two terms: $W_1 = 1$ and $W_2 = 2$ .
For any $n > 2$ , the $n^{\text{th}}$ term is given by the formula: $W_n = W_1 + W_2 + \dots + W_{n-2} + 2$
We compute each term sequentially from $W_3$ to $W_8$ and examine the resulting pattern.

Step 1 · Compute terms $W_3$ through $W_8$

Given $W_1 = 1$ and $W_2 = 2$ .

For $n = 3$ : $W_3 = W_1 + 2 = 1 + 2 = 3$

For $n = 4$ : $W_4 = W_1 + W_2 + 2 = 1 + 2 + 2 = 5$

For $n = 5$ : $W_5 = W_1 + W_2 + W_3 + 2 = 1 + 2 + 3 + 2 = 8$

For $n = 6$ : $W_6 = W_1 + W_2 + W_3 + W_4 + 2 = 1 + 2 + 3 + 5 + 2 = 13$

For $n = 7$ : $W_7 = W_1 + W_2 + W_3 + W_4 + W_5 + 2 = 1 + 2 + 3 + 5 + 8 + 2 = 21$

For $n = 8$ : $W_8 = W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + 2 = 1 + 2 + 3 + 5 + 8 + 13 + 2 = 34$

Step 2 · Identify the Pattern

The generated sequence of numbers is: $1, 2, 3, 5, 8, 13, 21, 34$

Notice that subtracting consecutive terms gives:

\begin{aligned} W_n - W_{n-1} &= (W_1 + W_2 + \dots + W_{n-2} + 2) - (W_1 + W_2 + \dots + W_{n-3} + 2) \\ &= W_{n-2} \end{aligned}

$\implies W_n = W_{n-1} + W_{n-2} \quad \text{for } n \ge 3$

Each term after the second is the sum of the preceding two terms. This is the Fibonacci sequence (starting from $1, 2$ ).

Answer

$W_1 = 1, W_2 = 2, W_3 = 3, W_4 = 5, W_5 = 8, W_6 = 13, W_7 = 21, W_8 = 34$ ; this is the Fibonacci sequence.

Common Mistakes

Index Off-by-One: Adding terms up to $W_{n-1}$ instead of $W_{n-2}$ . The formula specifically sums terms up to index $n-2$ before adding $2$ .
Forgetting the $+2$ constant: Omitting the final constant $+2$ when summing the preceding terms.

Question 15

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