6.06b - Fibonacci and geometric sequences

6.06b - Fibonacci and geometric sequences

How can a pattern grow when the amount added keeps changing? Learn to generate Fibonacci sequences, recognise quadratic patterns and identify geometric progressions by a constant positive rational multiplier.

Fibonacci sequences

Start with 1 and 1, then repeatedly add the two previous terms. This produces 1,1,2,3,5,8,13,…1,1,2,3,5,8,13,\ldots: each new value remembers two earlier values.

Fibonacci sequence

In a Fibonacci sequence, each term after the first two is the sum of the previous two. Both starting terms are needed. The familiar sequence begins 1, 1, but the same rule can be used with other starting terms.

Do not keep adding the same number: the pair being added moves along the list. To work backwards, subtract the known earlier term from the later total.

Worked example 1: going forwards

Question. A Fibonacci sequence starts 2,52,5. Find its next three terms.

Plan. Add the latest pair, then move that pair one place along.

2+5=75+7=127+12=19\begin{aligned} 2+5&=7 \\ 5+7&=12 \\ 7+12&=19 \end{aligned}

Answer. 2,5,7,12,192,5,7,12,19, so the next three terms are 7, 12 and 19.

Check. The differences 7−5=27-5=2, 12−7=512-7=5 and 19−12=719-12=7 recover the preceding terms.

Keep the same addition rule, but hide the first value. Reversing the sum finds the missing start.

Worked example 2: going backwards

Question. A Fibonacci sequence starts □,5,9,14,…\square,5,9,14,\ldots. Find the missing first term and the fifth term.

Plan. The first two terms sum to 9; the third and fourth sum to the fifth.

first term=9−5=4fifth term=9+14=23\begin{aligned} \text{first term}&=9-5=4 \\ \text{fifth term}&=9+14=23 \end{aligned}

Answer. The first term is 4 and the fifth is 23.

Check. 4+5=94+5=9, 5+9=145+9=14 and 9+14=239+14=23, so all overlapping pairs obey the rule.

Starting terms need not both be positive. Starting −2,3-2,3 gives 1,4,5,…1,4,5,\ldots; the addition rule still works. Starting 0,00,0 gives only zeros. The rule and starts determine the sequence, not a claim that every Fibonacci sequence must look like the familiar one.

Recognising quadratic sequences

Square numbers do not add the same amount each time, but their increases themselves follow a simple pattern.

For 1,4,9,16,251,4,9,16,25, the first differences are 3,5,7,93,5,7,9. The differences between those differences are 2,2,22,2,2. These are called second differences.

Quadratic sequence

A quadratic sequence has an nth-term rule containing a nonzero multiple of n2n^2, possibly with a multiple of nn and a constant added. Its second differences are constant and nonzero. An arithmetic sequence has constant first differences and zero second differences.

For a pattern described as quadratic, extend its first differences by the same second difference, then add to recover terms. You do not need to find its formula to do this.

Worked example 3: increasing differences

Question. The quadratic sequence 3,8,15,24,…3,8,15,24,\ldots continues by the same second-difference pattern. Find its next two terms.

Plan. Find the first differences, then extend them.

terms381524first differences579second differences22\begin{array}{c|rrrr} \text{terms}&3&8&15&24\\ \text{first differences}&&5&7&9\\ \text{second differences}&&&2&2 \end{array}

The next first differences are 9+2=119+2=11 and 11+2=1311+2=13.

24+11=35,35+13=4824+11=35,\qquad35+13=48

Answer. 35 and 48.

Check. The extended first differences 5,7,9,11,135,7,9,11,13 still increase by 2.

Now reverse the sign of the second difference. Constant second differences can be negative; the terms need not keep increasing.

Worked example 4: decreasing differences

Question. A quadratic sequence starts 12,15,16,15,…12,15,16,15,\ldots. Find its next two terms.

Plan. Use the signed differences, including the negative ones.

first differences:3,1,−1second differences:−2,−2next differences:−3,−5next terms:15−3=12,12−5=7\begin{aligned} \text{first differences}:&\quad3,1,-1 \\ \text{second differences}:&\quad-2,-2 \\ \text{next differences}:&\quad-3,-5 \\ \text{next terms}:&\quad15-3=12,\quad12-5=7 \end{aligned}

Answer. 12 and 7.

Check. From 16,15,12,716,15,12,7, the differences are −1,−3,−5-1,-3,-5, each 2 less than the preceding one.

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