1 = 1, 4 = 1 + 3, 9 = 1 + 3 + 5, 16 = 1 + 3 + 5 + 7, and so on. Each term in the square number sequence is the sum of the odd numbers up to that term. This interesting relationship between the odd numbers and square numbers can be represented by the diagram in the figure. Can you explain the relationship?
Step-by-step solution
Idea: Read the square diagram as a set of L-shaped bands, one inside another. Each band is one odd number of dots, and the bands together fill a square.
- Start in the corner with 1 dot: a 1 × 1 square. 1 = 12.½ mark
- Add an L-shaped band around it: 2 dots along one side and 2 along the other, sharing the corner dot, so 2 + 2 − 1 = 3 dots. Now we have a 2 × 2 square: 1 + 3 = 4 = 22.½ mark
- The next band has 3 + 3 − 1 = 5 dots and makes a 3 × 3 square: 1 + 3 + 5 = 9. The next has 7 dots: 1 + 3 + 5 + 7 = 16 = 42.½ mark
- In general, going from a (k − 1) × (k − 1) square to a k × k square adds k + k − 1 = 2k − 1 dots, the kth odd number.1 mark
- So the first n odd numbers fill an n × n square: 1 + 3 + 5 + … + (2n − 1) = n2. In the diagram, 8 bands give 1 + 3 + … + 15 = 64 = 82.½ mark
Check: First 8 odd numbers: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8 × 8 ✓.
Answer to write in the exam
1 dot = 1 × 1 square
Each L-shaped band from side k − 1 to side k adds k + k − 1 = 2k − 1 dots
Bands: 1, 3, 5, 7, … (the odd numbers)
First n bands fill an n × n square
∴ 1 + 3 + 5 + … + (2n − 1) = n2
Common mistakes that cost marks
- Counting a band as k + k = 2k dots. The corner dot is shared by both arms, so the band has 2k − 1 dots.
- Saying the sum of the first n odd numbers is (2n − 1)2. It is n2: the number of odd numbers added, squared.
- Describing the picture without linking it to the numbers. Say which band is which odd number and which square it completes.
How this can come in the exam
The sum 1 + 3 + 5 + … + 19 equals
- 81
- 100
- 190
- 361
Show answer
(B) 100
1 to 19 contains 10 odd numbers, so the sum is 102 = 100.
Assertion (A): 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.
Reason (R): The sum of the first n odd numbers is n2.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
There are 7 odd numbers, and 72 = 49. R is true and explains A.
Try one yourself
Without adding term by term, find 1 + 3 + 5 + … + 29.
Show answer
The odd numbers 1 to 29 are the first 15 odd numbers (since 2 × 15 − 1 = 29), so the sum is 152 = 225.
More questions like this
- 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.)
- 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?
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