If \(\mathrm{S(n)}\) is the sum of sequence 1, 2, 3, 4, ldotsn, in terms of n and \(\mathrm{S(n)}\), \(\mathrm{S(2n)}\)=?

1. Translate the problem requirements: \(\mathrm{S(n)}\) represents the sum \(1+2+3+\ldots+\mathrm{n}\). We need to find \(\mathrm{S(2n)}\) (the sum \(1+2+3+\ldots+2\mathrm{n}\)) expressed in terms of \(\mathrm{n}\) and \(\mathrm{S(n)}\).
2. Visualize the sequence extension: Break down \(\mathrm{S(2n)}\) into two parts - the first \(\mathrm{n}\) terms (which equals \(\mathrm{S(n)}\)) and the remaining terms from \((\mathrm{n}+1)\) to \(2\mathrm{n}\).
3. Calculate the sum of the additional terms: Find the sum of terms \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\) by factoring out \(\mathrm{n}\) from each term.
4. Combine the components: Add \(\mathrm{S(n)}\) to the sum of additional terms to get the final expression for \(\mathrm{S(2n)}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with in everyday terms. \(\mathrm{S(n)}\) represents adding up all the whole numbers from \(1\) to \(\mathrm{n}\). For example, if \(\mathrm{n} = 3\), then \(\mathrm{S(3)} = 1 + 2 + 3 = 6\). If \(\mathrm{n} = 5\), then \(\mathrm{S(5)} = 1 + 2 + 3 + 4 + 5 = 15\).

Now we need to find \(\mathrm{S(2n)}\), which means we're adding up all whole numbers from \(1\) to \(2\mathrm{n}\). So if \(\mathrm{n} = 3\), then \(\mathrm{S(2n)} = \mathrm{S(6)} = 1 + 2 + 3 + 4 + 5 + 6 = 21\). If \(\mathrm{n} = 5\), then \(\mathrm{S(2n)} = \mathrm{S(10)} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55\).

The question asks us to express \(\mathrm{S(2n)}\) in terms of both \(\mathrm{n}\) and \(\mathrm{S(n)}\).

Process Skill: TRANSLATE - Converting the sequence notation into concrete mathematical understanding

2. Visualize the sequence extension

Here's the key insight: when we go from \(\mathrm{S(n)}\) to \(\mathrm{S(2n)}\), we're essentially taking our original sequence and adding more terms to it.

Let's think about this step by step. \(\mathrm{S(2n)}\) means we're adding:
\(1 + 2 + 3 + \ldots + \mathrm{n} + (\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\)

Notice that we can split this into two parts:

• First part: \(1 + 2 + 3 + \ldots + \mathrm{n} = \mathrm{S(n)}\)
• Second part: \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\)

So \(\mathrm{S(2n)} = \mathrm{S(n)} + [(\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}]\)

Now we just need to figure out what that second part equals.

Process Skill: VISUALIZE - Breaking down the extended sequence into manageable components

3. Calculate the sum of the additional terms

Let's focus on finding the sum: \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\)

Here's a clever way to think about this. Each term in this sum can be written as:

• \((\mathrm{n}+1) = \mathrm{n} + 1\)
• \((\mathrm{n}+2) = \mathrm{n} + 2\)
• \((\mathrm{n}+3) = \mathrm{n} + 3\)
• ...
• \(2\mathrm{n} = \mathrm{n} + \mathrm{n}\)

So our sum becomes:
\((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n} = (\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + (\mathrm{n}+\mathrm{n})\)

We can factor out \(\mathrm{n}\) from each term:
\(= \mathrm{n} + \mathrm{n} + \ldots + \mathrm{n} + (1 + 2 + \ldots + \mathrm{n})\)

How many times does \(\mathrm{n}\) appear? Well, we have \(\mathrm{n}\) terms (from \((\mathrm{n}+1)\) to \(2\mathrm{n}\)), so we get \(\mathrm{n} \times \mathrm{n} = \mathrm{n}^2\).
And what's \((1 + 2 + \ldots + \mathrm{n})\)? That's just \(\mathrm{S(n)}\)!

Therefore: \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n} = \mathrm{n}^2 + \mathrm{S(n)}\)

4. Combine the components

Now we can put everything together:

\(\mathrm{S(2n)} = \mathrm{S(n)} + [(\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}]\)
\(\mathrm{S(2n)} = \mathrm{S(n)} + [\mathrm{n}^2 + \mathrm{S(n)}]\)
\(\mathrm{S(2n)} = \mathrm{S(n)} + \mathrm{n}^2 + \mathrm{S(n)}\)
\(\mathrm{S(2n)} = 2\mathrm{S(n)} + \mathrm{n}^2\)

Let's verify this with our earlier example where \(\mathrm{n} = 3\):

• \(\mathrm{S(3)} = 6\)
• \(\mathrm{S(6)} = 21\)
• Using our formula: \(\mathrm{S(6)} = 2\mathrm{S(3)} + 3^2 = 2(6) + 9 = 12 + 9 = 21 ✓\)

4. Final Answer

\(\mathrm{S(2n)} = 2\mathrm{S(n)} + \mathrm{n}^2\)

Looking at our answer choices, this matches choice (D) \(2\mathrm{S(n)}+\mathrm{n}^2\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what S(2n) represents

Students often think \(\mathrm{S(2n)}\) means "2 times \(\mathrm{S(n)}\)" rather than "the sum from 1 to \(2\mathrm{n}\)". This fundamental misinterpretation leads them to immediately select answer choice (A) without recognizing that \(\mathrm{S(2n)} = 1 + 2 + 3 + \ldots + 2\mathrm{n}\), which is a much larger sum than just doubling \(\mathrm{S(n)}\).

2. Failing to break down S(2n) systematically

Many students attempt to work directly with the formula \(\mathrm{S(n)} = \mathrm{n}(\mathrm{n}+1)/2\) without first understanding the structural relationship. They miss the key insight that \(\mathrm{S(2n)}\) can be split into \(\mathrm{S(n)}\) plus the additional terms from \((\mathrm{n}+1)\) to \(2\mathrm{n}\), leading to overly complicated algebraic manipulations.

3. Not recognizing the need to find the sum of consecutive integers

Students may identify that \(\mathrm{S(2n)} = \mathrm{S(n)} + \text{additional terms}\), but fail to realize they need to find the sum \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\). Instead, they might try to guess the relationship or use trial and error with the answer choices.

Errors while executing the approach

1. Incorrectly calculating the sum (n+1) + (n+2) + ... + 2n

When finding the sum of terms from \((\mathrm{n}+1)\) to \(2\mathrm{n}\), students often make arithmetic errors. They might incorrectly count how many terms there are (it's \(\mathrm{n}\) terms, not \(\mathrm{n}+1\) or \(\mathrm{n}-1\)), or make mistakes when factoring out \(\mathrm{n}\) from each term in the sequence.

2. Misapplying the arithmetic sequence formula

Some students try to use the arithmetic sequence sum formula directly on \((\mathrm{n}+1) + (\mathrm{n}+2) + \ldots + 2\mathrm{n}\) but make errors in identifying the first term, last term, or number of terms. For example, they might think there are \(2\mathrm{n} - \mathrm{n} + 1 = \mathrm{n} + 1\) terms instead of correctly identifying there are \(\mathrm{n}\) terms.

3. Algebraic manipulation errors when combining terms

Even when students correctly find that the additional sum equals \(\mathrm{n}^2 + \mathrm{S(n)}\), they may make errors when combining this with \(\mathrm{S(n)}\) to get the final answer. Common mistakes include writing \(\mathrm{S(n)} + \mathrm{n}^2 + \mathrm{S(n)} = \mathrm{S(n)} + \mathrm{n}^2\) or forgetting to combine like terms properly.

Errors while selecting the answer

1. Choosing answer choice (A) due to notation confusion

Students who misinterpret \(\mathrm{S(2n)}\) as "2 times \(\mathrm{S(n)}\)" will confidently select choice (A) \(2 \times \mathrm{S(n)}\) without verification, missing the crucial \(\mathrm{n}^2\) term that accounts for the additional sum from \((\mathrm{n}+1)\) to \(2\mathrm{n}\).

2. Selecting choice (E) due to incomplete reasoning

Some students correctly identify that there's an additional \(\mathrm{n}^2\) term but fail to account for the extra \(\mathrm{S(n)}\). They arrive at \(\mathrm{S(n)} + \mathrm{n}^2\) and select choice (E) \(\mathrm{S(n)}+2\mathrm{n}^2\), either by confusing \(\mathrm{n}^2\) with \(2\mathrm{n}^2\) or by incomplete analysis of the additional terms.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for n

Let's choose \(\mathrm{n} = 3\) (a small, manageable number that makes calculations straightforward)

Step 2: Calculate S(n) = S(3)

\(\mathrm{S(3)} = 1 + 2 + 3 = 6\)

Step 3: Calculate S(2n) = S(6)

\(\mathrm{S(6)} = 1 + 2 + 3 + 4 + 5 + 6 = 21\)

Step 4: Test each answer choice with our specific values

We need to find which expression equals \(21\) when \(\mathrm{n} = 3\) and \(\mathrm{S(n)} = 6\):

1. \(2 \times \mathrm{S(n)} = 2 \times 6 = 12\) ❌
2. \(\mathrm{n} \times \mathrm{S(n)} = 3 \times 6 = 18\) ❌
3. \(2\mathrm{n} \times \mathrm{S(n)} = 6 \times 6 = 36\) ❌
4. \(2\mathrm{S(n)} + \mathrm{n}^2 = 2 \times 6 + 3^2 = 12 + 9 = 21\) ✓
5. \(\mathrm{S(n)} + 2\mathrm{n}^2 = 6 + 2 \times 9 = 6 + 18 = 24\) ❌

Step 5: Verify with another value

Let's confirm with \(\mathrm{n} = 4\):

\(\mathrm{S(4)} = 1 + 2 + 3 + 4 = 10\)

\(\mathrm{S(8)} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36\)

Testing option (D): \(2\mathrm{S(n)} + \mathrm{n}^2 = 2 \times 10 + 4^2 = 20 + 16 = 36\) ✓

Conclusion: The answer is (D) \(2\mathrm{S(n)} + \mathrm{n}^2\)