Find all possible ways of expressing 100 as the sum of consecutive…

Question 42 of 51advanced🔍 AnalyzeLong Answer5 marks

Find all possible ways of expressing $100$ as the sum of consecutive natural numbers.

Correct Answer

100

18 + 19 + 20 + 21 + 22

9 + 10 + 11 + 12 + 13 + 14 + 15 + 16

Exercise: END-OF-CHAPTER EXERCISES | Q: 6 | (Page 22)

For More Understanding

Explanation

The textbook introduces the formula for the sum of the first $n$ natural numbers, $S_n = \frac{n(n+1)}{2}$ , and shows how to find the sum of consecutive numbers by subtracting sums. Using this, we set up an equation for the sum of consecutive numbers from $a+1$ to $b$ equaling $100$ .

Factoring leads to $(b-a)(b+a+1) = 200$ . Since the two factors must have different parities, we test the odd factors of $200$ ( $1$ , $5$ , $25$ ) to find valid natural number sequences.

Solution Steps

Step 1: Use the formula for the sum of consecutive natural numbers from $a+1$ to $b$ :
$S_b - S_a = \frac{b(b+1)}{2} - \frac{a(a+1)}{2} = 100$
Step 2: Multiply by 2 and rearrange:
$b(b+1) - a(a+1) = 200$
which simplifies to
$(b-a)(b+a+1) = 200$
Step 3: Let $x = b-a$ and $y = b+a+1$ . Since $y - x = 2a + 1$ is odd, $x$ and $y$ must have different parities (one odd, one even).
Step 4: The odd factors of $200$ are $1$ , $5$ , and $25$ . The valid pairs $(x, y)$ are $(1, 200)$ , $(5, 40)$ , and $(8, 25)$ .
Step 5: For $(x, y) = (1, 200)$ , $b=100$ and $a=99$ , giving the sequence:
$100$
Step 6: For $(x, y) = (5, 40)$ , $b=22$ and $a=17$ , giving the sequence:
$18 + 19 + 20 + 21 + 22$
Step 7: For $(x, y) = (8, 25)$ , $b=16$ and $a=8$ , giving the sequence:
$9 + 10 + 11 + 12 + 13 + 14 + 15 + 16$

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