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 . Here’s a step-by-step explanation of how AP works.
1. Definition and Formula
An arithmetic progression is defined by the sequence: where:
- is the first term of the sequence.
- is the common difference (the difference between any two consecutive terms).
2. Finding the th Term
To find the th term () of an AP, use the formula: This formula tells us that the th term is the first term plus times the common difference.
Example: In the sequence 3, 7, 11, 15, … the first term and the common difference . The 5th term would be:
3. Sum of the First Terms
The sum of the first terms of an AP is given by the formula: Alternatively, this can also be written as: where is the th term.
Example: For the sequence 3, 7, 11, 15, … the sum of the first 5 terms would be:
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 (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 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
- th term:
- Sum of the first terms:
Would you like a more detailed example or have any questions?
Practice Questions:
- Find the 10th term of the AP: 2, 5, 8, 11, …
- Calculate the sum of the first 20 terms of the AP: 4, 9, 14, 19, …
- If the first term of an AP is 7 and the 10th term is 52, find the common difference.
- Given an AP with first term 3 and common difference -2, what is the sum of the first 15 terms?
- 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 and the common difference 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
-
Suitable Grade Level
Grades 9-11
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