If S_x=8385, where S_x is the sum of the first x natural numbers, what is the value of x?
Answer and explanation
Correct answer: 129
For the sum of the first x natural numbers, use S_x=x(x+1)/2. We need x(x+1)/2=8385, so x(x+1)=16770. Testing the consecutive factors 129 and 130 gives 129×130=16770, and therefore S_129=129×130/2=8385. Hence x=129 and option B is correct. The neighboring choices fail because 128×129/2=8256, while 130×131/2=8515; thus neither produces the required sum. This also confirms uniqueness because the sum of positive natural numbers increases strictly as x increases.
Frequently asked questions
What is the correct answer to this question?
129
Why is this the correct answer?
For the sum of the first x natural numbers, use S_x=x(x+1)/2. We need x(x+1)/2=8385, so x(x+1)=16770. Testing the consecutive factors 129 and 130 gives 129×130=16770, and therefore S_129=129×130/2=8385. Hence x=129 and option B is correct. The neighboring choices fail because 128×129/2=8256, while 130×131/2=8515; thus neither produces the required sum. This also confirms uniqueness because the sum of positive natural numbers increases strictly as x increases.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Sum of first n natural numbers.
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