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Questions and Answers
What is the general formula for the $n^{th}$ term of an arithmetic progression?
What is the general formula for the $n^{th}$ term of an arithmetic progression?
The formula for the $n^{th}$ term is $a_n = a + (n - 1)d$ where $a$ is the first term and $d$ is the common difference.
How do you calculate the sum of the first $n$ terms in an arithmetic progression?
How do you calculate the sum of the first $n$ terms in an arithmetic progression?
The sum of the first $n$ terms is given by the formula $S_n = \frac{n}{2}(2a + (n - 1)d)$ or $S_n = \frac{n}{2}(a + a_n)$.
In an arithmetic progression, if the first term is 5 and the common difference is 3, what is the 10th term?
In an arithmetic progression, if the first term is 5 and the common difference is 3, what is the 10th term?
The 10th term is $a_{10} = 5 + (10 - 1) imes 3 = 32$.
What conditions must be met for a sequence to be classified as an arithmetic progression?
What conditions must be met for a sequence to be classified as an arithmetic progression?
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If the first term of an arithmetic progression is 2 and the 5th term is 12, what is the common difference?
If the first term of an arithmetic progression is 2 and the 5th term is 12, what is the common difference?
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Study Notes
General Concepts of Arithmetic Progression
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The general formula for the ( n^{th} ) term of an arithmetic progression (AP) is given by:
( a_n = a + (n-1)d )
where ( a ) is the first term, ( d ) is the common difference, and ( n ) is the term number. -
To calculate the sum of the first ( n ) terms ( (S_n) ) in an arithmetic progression, use:
( S_n = \frac{n}{2} [2a + (n-1)d] )
This formula is derived from the average of the first and last terms multiplied by the number of terms.
Specific Examples
- For an AP with a first term of 5 and a common difference of 3, the 10th term can be found by substituting into the formula:
( a_{10} = 5 + (10-1) \cdot 3 = 5 + 27 = 32 )
Characteristics of Arithmetic Progression
- A sequence is classified as an arithmetic progression if the difference between consecutive terms is constant (common difference, ( d )).
Finding Common Difference
- Given the first term of an arithmetic progression is 2 and the 5th term is 12, the common difference can be calculated:
( a_5 = a + 4d ) leads to
( 12 = 2 + 4d )
Solving gives ( 4d = 10 ), therefore ( d = 2.5 ).
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Description
Test your knowledge about arithmetic progressions with this quiz! You'll explore formulas for the nth term, the sum of terms, and specific calculations related to sequences. Perfect for students looking to deepen their understanding of this fundamental math topic.
