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The first five terms of a sequence are shown in the grid. Part a) Find the next term of the sequence.

To find the next term in a sequence, we always want to look for patterns. The first thing we want to check is does the sequence go up or down by the same
amount each time. To get from the first term in the sequence to the second term, we have to add
0.2. The same is true to get from the second term to the third term. In fact, it’s also true for getting from the third term to the fourth term and then
from the fourth term to the fifth term.

The rule for this sequence then is that we add 0.2 to the previous term. To find the next term in the sequence, we just need to add 0.2 to 1.2. We could do this mentally or using a column addition method and it’s equal to
1.4. The next term in the given sequence is 1.4.

The 𝑛th term of a different sequence is four 𝑛 squared plus 𝑛. Part b) Work out the sixth term of the sequence.

An 𝑛th term or 𝑛th term rule is a way of working out any term in the sequence based
on its term number. We’ve been asked to work out the sixth term of this sequence. So this means we’ll be substituting 𝑛 equals six into the given 𝑛th term rule. We need to be a little bit careful with this 𝑛th term rule as it begins with four 𝑛
squared.

If we remember the word BIDMAS or BODMAS, then we see that powers or indices come
before multiplication in the order of operations. So four 𝑛 squared means four multiplied by 𝑛 squared. We have to square 𝑛 first and then multiply it by four. A common mistake would be to multiply 𝑛 by four first and then square, which would
give a different and incorrect answer.

So we substitute six everywhere that 𝑛 appears in our 𝑛th term. And it gives four multiplied by six squared plus six. Six squared means six multiplied by six. And this is 36. So now, we have four multiplied by 36 plus six.

To work out what four multiplied by 36 is, we could use a column multiplication
method. First, we multiply six by four which is 24. So we write a four in the units column and carry the two into the tens column. Then, we multiply three by four which is 12 and adding on the two we’ve carried gives
14 or in fact 140 due to the place value of these numbers. So four multiplied by 36 is 144 and then adding the value six gives 150.

The sixth term of the sequence with 𝑛th term four 𝑛 squared plus 𝑛 is 150.