Alice is finishing her first term back at secondary school.
She has a maths test at the end of the week and has been working through some practice questions.
Her final question is about sequences.
The first four terms of a sequence are:
3, 9, 19, 33, …….
Help Alice find the 100th term of the sequence.
Answer:
3, 9, 19, 33, …….
The 1st difference between the terms is +6 then +10 then +14. Since these differences are not constant, we can look at the second difference. The difference between the 1st differences is a constant +4 each time, which shows us its an arithmetic sequence with an n2 nth term.
To find the nth term of this sequence we half the second difference of +4 which is 2. The start of our nth term expression is 2n2.
The sequence 2n2, where n is the number of terms (i.e. 1st term uses n=1, 2nd term uses n=2 etc), gives the sequence 2, 8, 18, 32 …
From this we can tell that the 2n2 sequence is 1 less than the sequence we are looking to find (3, 9, 19, 33…). So our nth term expression is 2n2 + 1, where n is the term number as before.
To find the 100th term we will use n=100:
2(100)2 + 1 = 20001