Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?

Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
Which best explains why Irving sets "The Adventure of the Mysterious Stranger" in a land of "masks and gondolas"?

A. The setting is symbolic of the idea that a life of quiet study is the ideal pursuit.
B. The setting is symbolic of the idea that innocence cannot be outgrown.
C. The setting is symbolic of the idea that ease and affluence are available to all.
D. The setting is symbolic of the idea that appearances can be deceiving.

Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
Read the lines from 'The Tide Rises, The Tide Falls.'

"Darkness settles on roofs and walls,
But the sea, the sea in darkness calls;"

The imagery in these lines evokes a sense of:
A. laziness
B. fear
C. mystery
D. despair

Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
Solve for x.

[tex]\[3x = 6x - 2\][/tex]

Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
[tex]\[1^3 + 3^3 + 5^3 + \cdots + (2n-1)^3 = n^2 \left(2n^2 - 1\right)\][/tex]



Answer :

To solve the problem of proving that:

[tex]\[1^3 + 3^3 + 5^3 + \cdots + (2n-1)^3 = n^2 \left(2n^2 - 1\right),\][/tex]

let's approach it step by step.

### Step 1: Understand the Sequence

The left-hand side (LHS) of the equation is the sum of the cubes of the first [tex]\(n\)[/tex] odd numbers:
[tex]\[ S = 1^3 + 3^3 + 5^3 + \cdots + (2n-1)^3. \][/tex]

### Step 2: Identify the General Terms

The [tex]\(k\)[/tex]-th odd number can be represented as [tex]\(2k - 1\)[/tex]. Therefore, the sequence is:
[tex]\[ (2 \times 1 - 1)^3, (2 \times 2 - 1)^3, (2 \times 3 - 1)^3, \ldots, (2 \times n - 1)^3. \][/tex]

### Step 3: Express the Sum

The sum can be expressed as:
[tex]\[ S = \sum_{k=1}^{n} (2k - 1)^3. \][/tex]

### Step 4: Expand the Cubes

Let's expand [tex]\((2k - 1)^3\)[/tex]:
[tex]\[ (2k - 1)^3 = 8k^3 - 12k^2 + 6k - 1. \][/tex]

### Step 5: Sum the Series

Summing these expanded terms from [tex]\(k = 1\)[/tex] to [tex]\(k = n\)[/tex]:
[tex]\[ S = \sum_{k=1}^{n} (8k^3 - 12k^2 + 6k - 1). \][/tex]

### Step 6: Separate the Summation

We can separate the sum into individual sums:
[tex]\[ S = 8 \sum_{k=1}^{n} k^3 - 12 \sum_{k=1}^{n} k^2 + 6 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1. \][/tex]

### Step 7: Use Summation Formulas

We use the known summation formulas:
[tex]\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \][/tex]
[tex]\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}, \][/tex]
[tex]\[ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2, \][/tex]
[tex]\[ \sum_{k=1}^{n} 1 = n. \][/tex]

### Step 8: Substitute

Substitute these into the separated sum:
[tex]\[ S = 8 \left( \frac{n(n+1)}{2} \right)^2 - 12 \left( \frac{n(n+1)(2n+1)}{6} \right) + 6 \left( \frac{n(n+1)}{2} \right) - n. \][/tex]

### Step 9: Simplify Each Term

Simplify each term separately:
1. [tex]\( 8 \left( \frac{n(n+1)}{2} \right)^2 = 8 \frac{n^2(n+1)^2}{4} = 2n^2(n+1)^2. \)[/tex]
2. [tex]\( -12 \left( \frac{n(n+1)(2n+1)}{6} \right) = -2n(n+1)(2n+1). \)[/tex]
3. [tex]\( 6 \left( \frac{n(n+1)}{2} \right) = 3n(n+1). \)[/tex]
4. [tex]\( -n. \)[/tex]

### Step 10: Combine the Simplified Results

Combine and simplify:
[tex]\[ S = 2n^2(n^2 + 2n + 1) - 2n(n^2 + 3n + 2) + 3n(n + 1) - n. \][/tex]

### Step 11: Factor and Simplify Further

Combine similar terms and factor the expression:
[tex]\[ S = 2n^2n^2 + 4n^2n + 2n^2 - 2n^3 - 6n^2 - 4n + 3n^2 + 3n - n. \][/tex]
[tex]\[ S = 2n^4 - 2n^2. \][/tex]

Therefore,
[tex]\[ S = 2n^4 - n^2 = n^2(2n^2 - 1). \][/tex]

### Conclusion

Hence, we have proved that:
[tex]\[ 1^3 + 3^3 + 5^3 + \cdots + (2n-1)^3 = n^2 \left( 2n^2 - 1 \right). \][/tex]