Sum of First n Terms of an AP

5.4 Sum of First n Terms of an AP

Description

Quick Overview

This section discusses how to calculate the sum of the first n terms of an arithmetic progression (AP).

Standard

The section introduces the concept of finding the sum of the first n terms of an arithmetic progression (AP) using formulas derived from the properties of APs. It explains the significance of the first term, the common difference, and employs historical examples to illustrate the concept.

Detailed

In this section, we explore the process of calculating the sum of the first n terms of an arithmetic progression (AP). An AP is defined as a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference (d). The sum of the first n terms can be represented by two formulas: \( S_n = \frac{n}{2} [2a + (n - 1)d] \) and \( S_n = \frac{n}{2} [a + l] \), where 'a' is the first term, 'l' is the last term, and 'n' is the number of terms. To illustrate the application of these formulas, the section presents a practical scenario involving the collection of money over the years and how to compute the total amount efficiently. Historical anecdotes about mathematicians like Gauss are shared to motivate students to appreciate the formulas intuitively. Additionally, examples and exercises are provided to reinforce the understanding of these formulas in diverse contexts.

Example 1: If the sum of the first 15 terms of the AP: 10, 7, 4, ... is 180, find the first term.
Solution: Here, \( a = 10, \ d = 7 - 10 = -3, \ n = 15 \).
We know that
\[ S_n = \frac{n}{2}[2a + (n-1)d] \]
Therefore,
\[ 180 = \frac{15}{2}[2(10) + (15-1)(-3)] \]
Thus, we have
\[ 180 = \frac{15}{2}[20 - 42] \]
So, the first term of the AP is 10.

Key Concepts

  • Sum of First n Terms: Formula to calculate the sum of the first n terms of an AP.

  • Arithmetic Structure: Understanding the structure and characteristics of an AP.

  • First and Last Terms: Roles of the first term and the last term in summation.

Memory Aids

🎵 Rhymes Time

  • To find S, it's nifty, just take a, add l, and then divide by two, real quick it's swell!

📖 Fascinating Stories

  • Imagine Shakila saves money for her daughter each birthday. The total grows like branches on a tree, spreading wide with each year. Adding them up is complicated, but using the magical sum formula makes it easy!

🧠 Other Memory Gems

  • Remember S = n/2 (first term + last term) as 'Silly Ninjas Paint Leafy Trees' to recall the elements: Sum, Number of terms, First and Last.

🎯 Super Acronyms

FORMULA for AP Sum

  • 'A Popular Hero' stands for 'a + l' (first term + last term) since it is central to finding our sum.

Examples

  • Finding the sum of the first 21 terms collected by Shakila amounts to a calculation simplifying the tedious addition process.

  • Using Gauss's technique of summing sequential numbers to derive a formula for calculating sums of terms in sequences.

Glossary of Terms

  • Term: Arithmetic Progression (AP)

    Definition:

    A sequence of numbers in which the difference between consecutive terms is constant.

  • Term: Common Difference (d)

    Definition:

    The fixed amount added to each term in an arithmetic progression.

  • Term: First Term (a)

    Definition:

    The initial term in an arithmetic progression.

  • Term: Nth Term (a_n)

    Definition:

    The term located at position n in an arithmetic progression.

  • Term: Sum of n Terms (S_n)

    Definition:

    The total value when summing the first n terms of an arithmetic progression.