For any positive integer z, SZ denotes the sum of the first z positive integers. For example S3 = 1+2 + 3 = 6. Which of the following expressions is correct?a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'E'. Can you explain this answer?

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For any positive integer z, S_Z denotes the sum of the first z positive integers. For example S₃ = 1+2 + 3 = 6. Which of the following expressions is correct?

a)
I only
b)
II only
c)
III only
d)
I and II only
e)
II and III only

Correct answer is option 'E'. Can you explain this answer?

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For any positive integer z, SZ denotes the sum of the first z positive...

Given:

For a positive integer z, S_z = 1 + 2 + 3 + . . . + z

To find: Which of the given 3 expressions is correct?

Approach:

Since the 3 expressions are independent of one another (that is, they do not talk about the same value or item), we will evaluate the expressions one by one, using the given definition of S_Z

Working Out:

Now, by definition, S₂₅ is the sum of the first 25 integers.

By definition,

So, 2S₁₀ + 15 = 2*55 + 15
- This value is clearly much smaller in magnitude than 25*13
Therefore, Expression 1 is not true.

Evaluating Expression II

Thus, we see that S₅₀ is indeed equal to 2S₂₅ + 625
Therefore, Expression II is correct

Evaluating Expression III
S₉₉ = 2S₅₀ + 2400

Note: You can calculate Expression III using the formula for the sum of an arithmetic sequence. The working out for this method will be similar to the working out shown above for Expressions I and II. Alternatively, you can solve as below. This is an alternate way of evaluating the given expressions. This way can also be used to evaluate Expressions I and II.

S₉₉ = 1 + 2 + 3 + . . . . + 99

= (1 + 2 + 3 + . . . + 99 + 100) – 100
= (1 + 2 + 3 + . . . + 50) + (51 + 52 + . . . + 100) - 100
= S₅₀ + {(50+1) + (50 + 2) . . . + (50 + 50)} – 100
= S₅₀ + 50*50 + (1 + 2 + 3 + . . . + 50) – 100
= S₅₀ + 2400 + S₅₀
= 2S₅₀+ 2400

Thus, Expression III is indeed correct

Conclusion
- So, we have determined that of the 3 given expressions, only Expressions II and III are correct.

Looking at the answer choices, we see that the correct answer is Option E

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For any positive integer z, SZ denotes the sum of the first z positive integers. For example S3 = 1+2 + 3 = 6. Which of the following expressions is correct?a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'E'. Can you explain this answer?

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For any positive integer z, SZ denotes the sum of the first z positive integers. For example S3 = 1+2 + 3 = 6. Which of the following expressions is correct?a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'E'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about For any positive integer z, SZ denotes the sum of the first z positive integers. For example S3 = 1+2 + 3 = 6. Which of the following expressions is correct?a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'E'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive integer z, SZ denotes the sum of the first z positive integers. For example S3 = 1+2 + 3 = 6. Which of the following expressions is correct?a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'E'. Can you explain this answer?.

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