Determine the fifth number in the following pattern, 80, 160, 480, 1920.
These four numbers are the first four numbers in a pattern. And, we’ve been asked to find out or determine the fifth number in the pattern. In order for us to be able to do this, we need to work out what the rule for the pattern is. So, let’s look at how the numbers change. Our first number is 80. And, the second number in the pattern is 160. Well, how do we get from 80 to 160? What could the rule be just by looking at these first two numbers?
Well, if it’s a one-step rule, then it’s either going to be plus 80, because 80 plus 80 equals 160, or multiplied by two, because 80 doubled is 160. The third number in our pattern is 480. The difference between 160 and 480 is 320. So, we can see that there isn’t an adding rule here. That would be very easy just to look at the numbers and think, “that’s a little bit larger; it must be multiplied by two,” and not actually try to work it out. But if we take time to think about it, we can see that multiplying by two is not the rule. We know that 16 doubled or 16 times two is 32. And so, 160 doubled, if that was the rule, would be 320. But, the third number in the sequence isn’t 320. It’s 480. What could the rule be?
What if the number that we’re multiplying by goes up each time? So, we might multiply it by two in the first instance, but for the second pair of numbers, we multiply it by three. Let’s see whether this works. What is 160 multiplied by three? Zero times three equals zero. Six threes are 18. And, one three is three plus the one underneath equals four. 160 multiplied by three does equal 480. It looks like we might have found our rule. We multiply the first number by two, the second number by three, and now we need to multiply 480 by four. Zero times four is zero. We know eight times two is 16. So, eight times four will be double 16 which is 32. And finally, four fours are 16 plus the three that we’ve exchanged equals 19.
As we expected, the fourth number in our sequence is 1920. So, now that we’ve worked out the rule, we can apply it to find the fifth number in the pattern. The rule is that we multiply by one more each time. Last time we multiplied by four, so now we need to multiply 1920 by five. Zero times five equals zero. Two fives are 10. Nine times five equals 45 plus the one that we’ve exchanged equals 46. And finally, one times five is five plus the four that we’ve exchanged equals nine.
In order to find the fifth number in the pattern, we had to first workout what the rule was. We found that the rule involved multiplication, and it changed every time. The number that we multiplied by increased by one. First, we multiplied by two, then by three, then by four. And to find the final number, the fifth number in the pattern, we had to multiply by five. So, multiplying 1920 by five gave us our answer. The fifth number in the pattern is 9600.