Math Problem Statement

Teach me Arithmetic progression

Solution

Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant value to the previous term. This constant is known as the common difference and is typically denoted by dd. Here’s a step-by-step explanation of how AP works.

1. Definition and Formula

An arithmetic progression is defined by the sequence: a,a+d,a+2d,a+3d,a, a + d, a + 2d, a + 3d, \ldots where:

  • aa is the first term of the sequence.
  • dd is the common difference (the difference between any two consecutive terms).

2. Finding the nnth Term

To find the nnth term (ana_n) of an AP, use the formula: an=a+(n1)da_n = a + (n - 1)d This formula tells us that the nnth term is the first term plus (n1)(n - 1) times the common difference.

Example: In the sequence 3, 7, 11, 15, … the first term a=3a = 3 and the common difference d=4d = 4. The 5th term would be: a5=3+(51)4=3+16=19a_5 = 3 + (5 - 1) \cdot 4 = 3 + 16 = 19

3. Sum of the First nn Terms

The sum of the first nn terms of an AP is given by the formula: Sn=n2(2a+(n1)d)S_n = \frac{n}{2} \cdot (2a + (n - 1)d) Alternatively, this can also be written as: Sn=n2(a+an)S_n = \frac{n}{2} \cdot (a + a_n) where ana_n is the nnth term.

Example: For the sequence 3, 7, 11, 15, … the sum of the first 5 terms would be: S5=52(23+(51)4)=52(6+16)=5222=55S_5 = \frac{5}{2} \cdot (2 \cdot 3 + (5 - 1) \cdot 4) = \frac{5}{2} \cdot (6 + 16) = \frac{5}{2} \cdot 22 = 55

4. Properties of Arithmetic Progressions

  • The difference between consecutive terms is always the same.
  • The sequence can increase or decrease based on the sign of dd (positive for increasing, negative for decreasing).
  • The average of the first and last terms in a finite AP gives the middle term (or the average of two middle terms if nn is even).

5. Applications of Arithmetic Progressions

APs are commonly found in real-life situations, like the seating arrangement of an auditorium, intervals of time, or pricing models where each item increases by a fixed amount.

Summary of Key Formulas

  • nnth term: an=a+(n1)da_n = a + (n - 1)d
  • Sum of the first nn terms: Sn=n2(2a+(n1)d)S_n = \frac{n}{2} \cdot (2a + (n - 1)d)

Would you like a more detailed example or have any questions?


Practice Questions:

  1. Find the 10th term of the AP: 2, 5, 8, 11, …
  2. Calculate the sum of the first 20 terms of the AP: 4, 9, 14, 19, …
  3. If the first term of an AP is 7 and the 10th term is 52, find the common difference.
  4. Given an AP with first term 3 and common difference -2, what is the sum of the first 15 terms?
  5. In an AP, if the 5th term is 20 and the 10th term is 35, find the common difference and first term.

Tip:

When working with APs, identifying the first term aa and the common difference dd is crucial for using the formulas correctly.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Progression
Sequences
Series

Formulas

nth term formula: a_n = a + (n - 1)d
Sum of the first n terms: S_n = (n/2) * (2a + (n - 1)d)

Theorems

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Suitable Grade Level

Grades 9-11