Find the sum of the first 47 terms of the following series, to the nearest integer.
12, comma, 16, comma, 20, comma, point, point, .
12,16,20,...

The given sequence is an arithmetic progression (AP) where the first term \( a = 12 \) and the common difference \( d = 16 - 12 = 4 \).

The formula for the sum of the first \( n \) terms of an arithmetic progression is:

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

Here:
- \( n = 47 \) (number of terms),
- \( a = 12 \) (first term),
- \( d = 4 \) (common difference).

Substituting these values into the formula:

\[
S_{47} = \frac{47}{2} \times (2 \cdot 12 + (47 - 1) \cdot 4)
\]

Let's calculate the sum.

\[
S_{47} = \frac{47}{2} \times (24 + 46 \cdot 4)
\]
\[
S_{47} = \frac{47}{2} \times (24 + 184)
\]
\[
S_{47} = \frac{47}{2} \times 208
\]
\[
S_{47} = 47 \times 104 = 4888
\]

Thus, the sum of the first 47 terms of the series is \( 4888 \).

Would you like more details or explanations? Here are some related questions:

1. How do you find the general term of an arithmetic series?
2. What is the difference between arithmetic and geometric sequences?
3. How do you derive the formula for the sum of an arithmetic series?
4. How can you find the sum if you know only the first and last term?
5. What are other applications of arithmetic progressions in real life?

**Tip:** Always check if a sequence follows a pattern like arithmetic or geometric before applying formulas.

Find the sum of the first 47 terms of the following series: 12, 16, 20, ...

Find the sum of the first 47 terms of the arithmetic progression: 12, 16, 20,... with a common difference of 4. This problem involves the sum formula for arithmetic progressions and is suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation