Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.)
Step-by-step solution
Idea: The sequence goes up by 3 each time. The new sequence is made of running totals: each new term = previous running total + the next term of 1, 4, 7, 10, …
- Differences: 4 − 1 = 3, 7 − 4 = 3, … so add 3 each time.½ mark
- Next four terms: 13 + 3 = 16, 19, 22, 25. (We will also need the 10th term: 28.)½ mark
- Running totals: 1; 1 + 4 = 5; 5 + 7 = 12; 12 + 10 = 22; 22 + 13 = 35.1 mark
- Continue: 35 + 16 = 51; 51 + 19 = 70; 70 + 22 = 92; 92 + 25 = 117; 117 + 28 = 145.½ mark
- First 10 terms of the new sequence: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.½ mark
Check: The 10 terms 1, 4, …, 28 pair up from the ends: (1 + 28) + (4 + 25) + … = 5 pairs × 29 = 145 ✓.
Answer to write in the exam
Common difference = 3
Next four terms: 16, 19, 22, 25
Running sums: 1, 1 + 4 = 5, 5 + 7 = 12, 12 + 10 = 22, 22 + 13 = 35
35 + 16 = 51, 51 + 19 = 70, 70 + 22 = 92, 92 + 25 = 117, 117 + 28 = 145
∴ 1, 5, 12, 22, 35, 51, 70, 92, 117, 145
Common mistakes that cost marks
- Starting each running sum again from the original term instead of the previous total (e.g. writing 4 + 7 = 11 instead of 1 + 4 + 7 = 12).
- Stopping at the 9th term because only 9 terms of 1, 4, 7, … were written. The 10th running sum needs the 10th term, 28.
- Adding 3 to the running sums (1, 4, 7, …) instead of adding the next term of the original sequence.
How this can come in the exam
For the sequence 2, 5, 8, 11, …, what is the sum of its first four terms?
- 11
- 24
- 26
- 15
Show answer
(C) 26
2 + 5 + 8 + 11 = 26.
Write the first five running sums of 3, 7, 11, 15, 19, …
Show answer
3; 3 + 7 = 10; 10 + 11 = 21; 21 + 15 = 36; 36 + 19 = 55. So 3, 10, 21, 36, 55.Try one yourself
Write the next three terms of 2, 7, 12, 17, … and the first six running sums.
Show answer
Next terms: 22, 27, 32. Running sums: 2, 9, 21, 38, 60, 87.
More questions like this
- Can you write t5, t6, t7 and t8 for the sequence of triangular numbers?
- Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.
- Consider the expression un = 2n − 1. This states that the nth term of the sequence is given by the rule 2n − 1.
- Why is it useful to have an explicit formula for the nth term of a sequence?
- Using the explicit rule un = 2n − 1, find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.
All Sequences and progressions questions · All maths questions