The first three terms of a sequence are shown below. Brandon believes he knows the first six terms of the sequence, and he writes down the following numbers. a) Describe the rules that Brandon used to find the fourth, fifth, and sixth terms of his sequence.
It’s always sensible when presented with a sequence to check whether there is a common difference or a pattern in the differences between the terms of the sequence. To do this, we find the differences between the consecutive terms.
Five minus one is four, 10 minus five is five, 50 minus 10 is 40, and 100 minus 50 is 50. There is a slight pattern here, though not enough to form a simple rule.
Instead, let’s see if we can find another relationship between the terms by dividing consecutive terms. Five divided by one is five, 10 divided by five is two, 50 divided by 10 is five, and 100 divided by 50 is two again. And in fact, 500 divided by 100 is five again.
What’s happening here is that Brandon is alternating by multiplying by five then two then five, and so on. To get to the fourth term, he multiplied the third by five. To get to the fifth term, he multiplied the fourth by two. And to get to the sixth term, he multiplied the fifth by five again.
Part b) Write down the seventh term of Brandon’s sequence.
Since he multiplied by five to get to the sixth term, he would now multiply by two again. 500 multiplied by two is 1000, so the seventh term in his sequence is 1000.
Callum believes the first six terms of the sequence are different. He writes down the following numbers. c) Write down the seventh and eighth terms of Callum’s sequence.
Once again, let’s see what the differences are between the terms. Five minus one is four, 10 minus five is five, 16 minus 10 is six, and then we get a difference of seven and a difference of eight. To get to the seventh term in the sequence then, we will add nine to 31. 31 plus nine is 40. Then to get to the eighth term, we’ll add 10. 40 plus 10 is 50. The seventh and eighth terms in this sequence are 40 and 50.