Week 319 (20th October 2023)

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. 


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 nnth 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