Which of the following statements is sufficient to answer the question: “Find the sum of the first 10 numbers in the series”? Statement I: The first number of the series is 2 and the tenth number is 20. Statement II: The common difference of the series is 2.

Introduction / Context:
This is a data sufficiency problem involving an arithmetic series. You are not asked to compute the actual numerical sum, but to determine which given information is enough to calculate that sum. Understanding the structure of arithmetic progressions and how to use the formula for the sum of the first n terms is essential to solve this type of question efficiently.

Given Data / Assumptions:

• Question: Find the sum of the first 10 numbers in the series.
• Statement I: The first number is 2 and the tenth number is 20.
• Statement II: The common difference of the series is 2.
• We treat the series as an arithmetic progression unless stated otherwise, which is the standard interpretation in such questions.

Concept / Approach:
For an arithmetic progression, the sum of the first n terms is given by S_n = n * (first term + nth term) / 2. To find the sum of the first 10 terms, you only need the first term and the tenth term. If you are given the first term and the common difference, you can also compute the tenth term and hence the sum. However, being given only the common difference without any term value is not enough, because the entire series could be shifted up or down by a constant and still have the same common difference.

Step-by-Step Solution:
Step 1: From Statement I, we know the first term a₁ = 2 and the tenth term a₁₀ = 20.Step 2: For an arithmetic progression, the sum of the first 10 terms is S₁₀ = 10 * (a₁ + a₁₀) / 2.Step 3: Because both a₁ and a₁₀ are given in Statement I, we can directly compute S₁₀. Hence Statement I alone is sufficient.Step 4: From Statement II, we only know that the common difference d = 2. We do not know the first term or any specific term value.Step 5: Many different arithmetic series share the same common difference of 2. For example, the series could be 2, 4, 6, ... or 5, 7, 9, ... or negative values. Each would have a different sum for the first 10 terms.Step 6: Therefore Statement II alone is not sufficient to determine the sum.Step 7: Since Statement I is already sufficient by itself, combining it with Statement II is unnecessary.

Verification / Alternative check:
Using Statement I, you can also derive the common difference if needed: d = (a₁₀ - a₁) / 9 = (20 - 2) / 9 = 2. This confirms that the series is indeed arithmetic with d = 2. Then S₁₀ can be computed consistently. On the other hand, if you start only with d = 2 and no term values, there is an infinite number of possible series and sums, confirming that Statement II alone is insufficient.

Why Other Options Are Wrong:

• Option a: Only Statement II is sufficient is incorrect because knowing only the common difference does not fix the series or the sum.
• Option c: Both statements together are sufficient suggests that neither alone works, which is not true since Statement I is already enough.
• Option d: Either statement alone is sufficient is wrong because Statement II alone is not sufficient.
• Option e: Neither statement is sufficient contradicts the clear sufficiency of Statement I.

Common Pitfalls:

• Confusing what is needed to compute the sum of an arithmetic series and thinking that the common difference alone is enough.
• Misreading Statement I and not recognising that the tenth number given is exactly the term required in the sum formula.
• Assuming that every series with a fixed common difference must start from a fixed first term, which is not true.

Final Answer:
Only statement I is sufficient to determine the sum of the first 10 numbers in the series.