Patterns in Mathematics | IT

Question 3

By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?

Question diagram 1
Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • Adding consecutive odd numbers starting from 11 forms square arrays of dots:
    • 1=1×1=121 = 1 \times 1 = 1^2
    • 1+3=4=2×2=221 + 3 = 4 = 2 \times 2 = 2^2
    • 1+3+5=9=3×3=321 + 3 + 5 = 9 = 3 \times 3 = 3^2
  • In general, the sum of the first nn odd numbers equals n×n=n2n \times n = n^2.
  • To find the sum of the first 1010 odd numbers, we calculate 10×1010 \times 10.

Step 1 · Identify the Pattern

Diagram 1

Observe the sum of consecutive odd numbers starting from 11

1=1×1=11 = 1 \times 1 = 1 1+3=4=2×2\begin{aligned} 1 + 3 &= 4 \\ &= 2 \times 2 \end{aligned} 1+3+5=9=3×3\begin{aligned} 1 + 3 + 5 &= 9 \\ &= 3 \times 3 \end{aligned} 1+3+5+7=16=4×4\begin{aligned} 1 + 3 + 5 + 7 &= 16 \\ &= 4 \times 4 \end{aligned}

Thus, the sum of the first nn odd numbers is n×nn \times n.

Step 2 · Calculate the Sum of the First 10 Odd Numbers

The first 1010 odd numbers are 1,3,5,7,9,11,13,15,17,191, 3, 5, 7, 9, 11, 13, 15, 17, 19.

Using the pattern for n=10n = 10

1+3+5+7+9+11+13+15+17+19=10×10=100\begin{aligned} 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 &= 10 \times 10 \\ &= 100 \end{aligned}
Answer

100

Common Mistakes
  • Adding numbers up to 10 instead of the first 10 odd numbers: Calculating 1+3+5+7+9=251 + 3 + 5 + 7 + 9 = 25 (which is only the first 55 odd numbers) instead of adding the first 1010 odd numbers (which go up to 1919).
  • Arithmetic addition errors: Adding the ten numbers manually one-by-one rather than using the square pattern n×nn \times n, increasing the chance of calculation mistakes.

More questions in IT

Q1

Why does this happen? Do you think it will happen forever?

Q2

How can we partition the dots in a square grid into odd numbers of dots: 1,3,5,7,…1, 3, 5, 7, \dots ?

Q3

By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?

Q4

Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?

Q5

Can you find a similar pictorial explanation?

← Back to Patterns in Mathematics