Flash Cards: Arithmetic Progressions
Arithmetic Progressions - Flash Cards
1/19
What is an Arithmetic Progression (AP)?
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An Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term.
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2/19
What is the common difference in an AP?
2/19
The common difference (d) is the fixed amount added to each term to get the next term in an Arithmetic Progression.
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3/19
What is the general form of an AP?
3/19
The general form of an Arithmetic Progression can be expressed as a, a+d, a+2d, a+3d, ..., where 'a' is the first term and 'd' is the common difference.
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4/19
How can you identify if a sequence is an AP?
4/19
A sequence is an AP if the difference between consecutive terms is constant, i.e., ak+1 - ak = d.
5/19
How do you find the nth term of an AP?
5/19
The nth term (an) of an AP can be calculated using the formula: an = a + (n-1)d.
6/19
What is the formula for the sum of the first n terms of an AP?
6/19
The sum of the first n terms (Sn) of an AP is given by the formula: Sn = n/2 * (2a + (n-1)d).
7/19
Give an example of an AP.
7/19
An example of an AP is: 2, 5, 8, 11, ..., where the first term a = 2 and the common difference d = 3.
8/19
What is the difference between finite and infinite AP?
8/19
A finite AP has a specific number of terms, while an infinite AP continues indefinitely without a last term.
9/19
How do you find the common difference in a given AP?
9/19
To find the common difference d, subtract any term from the term that follows it, e.g., d = a2 - a1.
10/19
What is a constant sequence in AP?
10/19
In a constant sequence where all terms are equal, d = 0 and the sequence is still considered an AP.
11/19
Can an AP be expressed as a recurrence relation?
11/19
Yes, an AP can be expressed as: an = an-1 + d, with a1 defined as the initial term.
12/19
Where can you find real-life applications of AP?
12/19
Real-life applications of AP include calculating salaries, distances covered, or any situation with uniform increments.
13/19
Is the sequence 2, 4, 8, 16 an AP?
13/19
No, because the common difference is not constant; this is a geometric progression instead.
14/19
How do you write the first four terms given a = 3 and d = 2?
14/19
The first four terms are: 3, 5, 7, 9.
15/19
What does a negative common difference indicate?
15/19
A negative common difference indicates the terms of the AP decrease as you progress through the sequence.
16/19
How do you find the sum if d is negative?
16/19
The formula for the sum of the first n terms remains the same, but the individual terms will decrease.
17/19
How do you determine the number of terms in a finite AP?
17/19
You can determine the number of terms by using the formula n = (last term - first term)/d + 1.
18/19
What is a common mistake when identifying AP?
18/19
A common mistake is considering a sequence an AP if the terms differ in a non-uniform manner.
19/19
How can you construct an AP from a story problem?
19/19
Identify the initial amount and the rate of change per term, then use these to establish the first term and the common difference.
