Math Problem Statement

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,...

Solution

The given sequence is an arithmetic progression (AP) where the first term a=12a = 12 and the common difference d=1612=4d = 16 - 12 = 4.

The formula for the sum of the first nn terms of an arithmetic progression is:

Sn=n2×(2a+(n1)d)S_n = \frac{n}{2} \times (2a + (n - 1) \cdot d)

Here:

  • n=47n = 47 (number of terms),
  • a=12a = 12 (first term),
  • d=4d = 4 (common difference).

Substituting these values into the formula:

S47=472×(212+(471)4)S_{47} = \frac{47}{2} \times (2 \cdot 12 + (47 - 1) \cdot 4)

Let's calculate the sum.

S47=472×(24+464)S_{47} = \frac{47}{2} \times (24 + 46 \cdot 4) S47=472×(24+184)S_{47} = \frac{47}{2} \times (24 + 184) S47=472×208S_{47} = \frac{47}{2} \times 208 S47=47×104=4888S_{47} = 47 \times 104 = 4888

Thus, the sum of the first 47 terms of the series is 48884888.

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.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Progression
Sequences and Series

Formulas

Sum of Arithmetic Progression: S_n = (n/2) × (2a + (n - 1) × d)

Theorems

Arithmetic Progression Sum Formula

Suitable Grade Level

Grades 9-12