Question

Use S=n 2to find the sum of 1+3+5+…+301.

A. 22,801
B. 22,802
C. 22,500
D. 22,800

Answer 1

To find the sum of the arithmetic series 1 + 3 + 5 + ··· + 301, we can use the formula for the sum of an arithmetic series. The correct sum of 1 + 3 + 5 + ··· + 301 is 22,801. The answer is A. 22,801.

Step-by-step explanation:

To find the sum of the arithmetic series 1 + 3 + 5 + ··· + 301, we can use the formula for the sum of an arithmetic series:

Sₙ =
`(n)/(2)`[2a + (n-1)d]

where:

- Sₙ is the sum of the series,

- n is the number of terms,

- a is the first term, and

- d is the common difference between terms.

In this case:

- n is the number of terms,

- a is the first term (1),

- d is the common difference (2).

1. Identify the values:

- n (number of terms) = ?

- a (first term) = 1

- d (common difference) = 2

2. Determine n:

The nth term can be found using the formula for the nth term of an arithmetic sequence:

aₙ = a + (n-1)d

For aₙ = 301, a = 1, and d = 2:

301 = 1 + (n-1) · 2

Solve for n.

3. Plug the values into the formula for Sₙ:

Sₙ =
`(n)/(2)` [2a + (n-1)d]

Now, let's go through these steps:

1. Find n:

301 = 1 + (n-1) · 2

300 = (n-1) · 2

n-1 = 150

n = 151

2. Use the formula for Sₙ:

S₁₅₁ =
`(151)/(2)` [2 · 1 + (151 - 1) · 2]

3. Calculate S₁₅₁.

S₁₅₁ =
`(151)/(2)` [2 + (150) · 2]

S₁₅₁ =
`(151)/(2)` [2 + 300]

S₁₅₁ =
`(151)/(2)` [302]

S₁₅₁ = 151 · 151

S₁₅₁ = 22,801

So, the correct sum of 1 + 3 + 5 + ··· + 301 is 22,801.