nth Term of an AP

5.3 nth Term of an AP

Description

Quick Overview

This section explains how to compute the nth term of an Arithmetic Progression (AP) and provides various examples illustrating this concept.

Standard

In this section, we explore the concept of the nth term in an Arithmetic Progression (AP). By analyzing Reena's salary increment example, we derive the formula for the nth term of an AP, a_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. The section also includes several examples showcasing how to apply this formula to determine specific terms in various arithmetic sequences.

Detailed

nth Term of an AP

In this section, we delve into how to find the nth term of an Arithmetic Progression (AP). An AP is a sequence where each term after the first is formed by adding a constant, known as the common difference (d), to the previous term.

Key Points:

  • Understanding the nth term: The nth term in an AP can be found using the formula: a_n = a + (n - 1)d, where:
  • a is the first term,
  • d is the common difference.
  • Example Application: Consider the case of Reena, who starts with a monthly salary of ₹8000 and receives an annual increment of ₹500. To find her salary in the 5th year, we can compute:
  • Salary for the 5th year = a + (5-1)d = ₹8000 + 4 × ₹500 = ₹10000.
  • Examples and Practice: Further examples illustrate finding specific terms in various sequences, checking if a number is part of an AP, and reverse calculations to derive values like the first term or common difference.

This formula and understanding are crucial in various applications such as financial forecasting, pattern recognition, and more. Mastery of these concepts allows for efficient resolution of complex arithmetic-related problems.

Key Concepts

  • nth Term: Formula for nth term is a_n = a + (n - 1)d.

  • Common Difference: The difference, d, that remains constant between terms.

  • First Term: The starting point of the AP denoted as 'a'.

  • The Importance of APs: Used in practical financial planning and budgeting.

Memory Aids

🎵 Rhymes Time

  • To find the nth term, just take a, add d times (n minus one), that’s how it’s done.

📖 Fascinating Stories

  • Imagine a bank where you're depositing a fixed amount each month; your balance grows steadily, just like terms in an AP!

🧠 Other Memory Gems

  • Remember A for Arithmetic, D for Difference to find the nth term!

🎯 Super Acronyms

AP = (A + (n-1)D) for finding terms swiftly.

Examples

  • If a = 2 and d = 5, then 10th term = 2 + (10 - 1) * 5 = 47.

  • For the AP 4, 7, 10, find the 7th term: a = 4, d = 3; hence a_7 = 4 + (7-1)*3 = 22.

Glossary of Terms

  • Term: Arithmetic Progression (AP)

    Definition:

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

  • Term: Common Difference (d)

    Definition:

    The fixed amount added to each term in an Arithmetic Progression to obtain the next term.

  • Term: First Term (a)

    Definition:

    The initial term in an Arithmetic Progression.

  • Term: nth Term (a_n)

    Definition:

    The term in the sequence of an AP that corresponds to the index n.

Example 2

In a flower bed, there are 30 tulip plants in the first row, 28 in the second, 26 in the third, and so on. There are 10 plants in the last row. How many rows are there in the flower bed?

Solution: The number of tulip plants in the 1st, 2nd, 3rd, ..., rows are:
\[ 30, 28, 26, \ldots \]
It forms an AP (Arithmetic Progression) (Why?). Let the number of rows in the flower bed be \( n \).

Then,
\[ a = 30, \, d = 28 - 30 = -2, \, a_n = 10 \]
As,
\[ a_n = a + (n - 1) d \]
We have,
\[ 10 = 30 + (n - 1)(-2) \]
\[ 10 = 30 - 2(n - 1) \]
\[ 10 = 30 - 2n + 2 \]
\[ 2n = 30 + 2 - 10 \]
\[ 2n = 22 \]
\[ n = 11 \]

i.e., So there are 11 rows in the flower bed.