Patterns in Mathematics | IT

Question 4

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?

Question diagram 1
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Solution
Understand the Question
  • Adding consecutive odd numbers starting from 11 forms square numbers of increasing sizes.
  • The sum of the first nn odd numbers is always equal to n2n^2.
  • To find the sum of the first 100100 odd numbers, we apply this rule with n=100n = 100.

Step 1 · Observe the Geometric Pattern

Diagram 1

Observing the sum of the first few odd numbers:

Sum of first 1 odd number=1=12\text{Sum of first } 1 \text{ odd number} = 1 = 1^2

Sum of first 2 odd numbers=1+3=4=22\text{Sum of first } 2 \text{ odd numbers} = 1 + 3 = 4 = 2^2

Sum of first 3 odd numbers=1+3+5=9=32\text{Sum of first } 3 \text{ odd numbers} = 1 + 3 + 5 = 9 = 3^2

Sum of first 4 odd numbers=1+3+5+7=16=42\text{Sum of first } 4 \text{ odd numbers} = 1 + 3 + 5 + 7 = 16 = 4^2

Step 2 · Establish the General Rule

From the pattern, the sum of the first nn odd numbers is:

Sum=n2\text{Sum} = n^2

Step 3 · Calculate the Sum of First 100 Odd Numbers

For the first 100100 odd numbers, substitute n=100n = 100:

1002=100×100=10,000\begin{aligned} 100^2 &= 100 \times 100 \\ &= 10{,}000 \end{aligned}
Answer

10,00010{,}000

Common Mistakes
  • Adding Manually: Attempting to manually add all 100100 numbers instead of using the formula Sum=n2\text{Sum} = n^2.
  • Confusing the nn-th Term with the Sum: The 100th100^{\text{th}} odd number is 2(100)−1=1992(100) - 1 = 199, while the sum of the first 100100 odd numbers is 1002=10,000100^2 = 10{,}000.

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?

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