# Video: Pack 1 • Paper 3 • Question 17

Pack 1 • Paper 3 • Question 17

02:47

### Video Transcript

Below is a sequence containing six terms: six, 13, 26, 45, 70, and 101. Write an expression in terms of 𝑛 for the 𝑛th term of this sequence.

A good place to start is to look for the first difference of the sequence. Notice how the difference changes each time. This means this is not an arithmetic sequence. However, the second difference is plus six each time. This means it’s a quadratic sequence, 𝑛 squared.

To find the coefficient of 𝑛 squared, which is the number in front of 𝑛 squared, we halve the second difference. Half of six is three. So our coefficient of 𝑛 squared is three. Next, we should write the sequence three 𝑛 squared out above our original sequence.

When 𝑛 is one, three 𝑛 squared is three multiplied by one squared, which is three. When 𝑛 is two, three 𝑛 squared is three multiplied by two squared, which is 12. Remember, we must work out the index or the power before we multiply. When 𝑛 is three, three 𝑛 squared is three multiplied by three squared, or 27. And when 𝑛 is four, three 𝑛 squared is three multiplied by four squared, which is 48. We don’t need to write out the whole sequence. Three or four terms is sufficient to help us identify a pattern.

Now let’s see what we need to do to get from our sequence of three 𝑛 squared to the sequence we were given in the question. For the first term, we add three. To get from 12 to 13, we add one. To get from 27 to 26, we subtract one. And to get from 48 to 45, we subtract three.

Take a look at this sequence. You might have noticed we’ve created a linear sequence. This sequence has a common difference of negative two. So its 𝑛th term starts with negative two 𝑛. The negative two times table is negative two, negative four, negative six, and negative eight.

To get from this sequence to ours, we add five each time. So the 𝑛th term for this new sequence that we created is negative two 𝑛 plus five. Putting this all together, we get that the 𝑛th term of the quadratic sequence is three 𝑛 squared minus two 𝑛 plus five.