Question 14 of 51beginner🔧 ApplyEssay2 marks

Can you use this formula to find S20S_{20}, S50S_{50} or S1000S_{1000}?

Correct Answer

Yes, we can use the formula to find these sums. The formula for the sum of the first n natural numbers is given by Sn=n(n+1)2S_n = \frac{n(n+1)}{2}. By substituting the value of n into this formula, we can easily calculate the sum without adding all the numbers individually.

To find S20S_{20}, substitute n=20: S20=20×212=210S_{20} = \frac{20 \times 21}{2} = 210. To find S50S_{50}, substitute n=50: S50=50×512=1275S_{50} = \frac{50 \times 51}{2} = 1275.

Similarly, for S1000S_{1000}, substitute n=1000: S1000=1000×10012=500500S_{1000} = \frac{1000 \times 1001}{2} = 500500. Thus, the formula allows us to find the sum for any value of n efficiently.

Exercise: Think and Reflect | Q: 1 | (Chapter: 12)
For More Understanding

Explanation

The textbook context establishes the general formula for the sum of the first n natural numbers as Sn=n(n+1)2S_n = \frac{n(n+1)}{2}. The 'Think and Reflect' question prompts students to apply this formula for larger values of n (20, 50, 1000) to demonstrate its utility over manual addition. By substituting these values into the formula, students can quickly compute the respective sums, reinforcing the concept that the formula works for any integer n.

Solution Steps

  1. Step 1: State the formula for the sum of the first n natural numbers: Sn=n(n+1)2S_n = \frac{n(n+1)}{2}.

  2. Step 2: Calculate S20S_{20} by substituting n=20: S20=20×212=210S_{20} = \frac{20 \times 21}{2} = 210.

  3. Step 3: Calculate S50S_{50} by substituting n=50: S50=50×512=1275S_{50} = \frac{50 \times 51}{2} = 1275.

  4. Step 4: Calculate S1000S_{1000} by substituting n=1000: S1000=1000×10012=500500S_{1000} = \frac{1000 \times 1001}{2} = 500500.

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