Question

The sum of the first 20 terms of the arithmetic progression 7, 10, 13, ... is:

• A. 470
• B. 710
• C. 670
• D. 770

Hint:

Formula: \( S_n = \frac{n}{2} [2a + (n-1)d] \) is your go-to for sum of an arithmetic series.

Answer 1

The Correct Option is B

To find the sum of the first 20 terms of the arithmetic progression (AP) given by the sequence 7, 10, 13, ..., we use the formula for the sum of the first n terms of an AP:

Sum = \( \frac{n}{2} \times (\text{First term} + \text{Last term}) \)

where
First term \( a = 7 \)
Common difference \( d = 10 - 7 = 3 \)

The n-th term of an AP can be calculated using the formula:

\( a_n = a + (n-1) \cdot d \)

For the 20th term (n=20):

\( a_{20} = 7 + (20-1) \cdot 3 \)

\( a_{20} = 7 + 57 = 64 \)

Now, use the sum formula:

Sum = \( \frac{20}{2} \times (7 + 64) \)

\( \text{Sum} = 10 \times 71 = 710 \)

Thus, the correct answer is 710.

Answer 2

To find the sum of the first 20 terms of the arithmetic progression (AP) defined by the sequence 7, 10, 13, ..., we apply the formula for the sum of the first \(n\) terms of an AP:

\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]

where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(n\) is the number of terms,
• \(a\) is the first term,
• \(d\) is the common difference.

For this sequence:

• \(a = 7\) (the first term),
• \(d = 3\) (common difference, as \(10 - 7 = 3\)),
• \(n = 20\) (since we need the sum of the first 20 terms).

Substitute these values into the formula:

\[ S_{20} = \frac{20}{2} \times [2 \times 7 + (20-1) \times 3] \]

Calculating step-by-step:

• \( \frac{20}{2} = 10 \)
• \( 2 \times 7 = 14 \)
• \( (20-1) \times 3 = 19 \times 3 = 57 \)
• \( 14 + 57 = 71 \)
• \( S_{20} = 10 \times 71 = 710 \)

Thus, the sum of the first 20 terms is 710.