Arithmetic Progressions – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Arithmetic Progressions chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
General term of an AP: a_n = a + (n - 1)d
a_n is the nth term, a is the first term, d is the common difference, and n is the term number. This formula allows you to find any term in the arithmetic progression.
Sum of first n terms (S_n) of an AP: S_n = n/2 (2a + (n - 1)d)
S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms. This helps calculate the total of the first n terms quickly.
Sum of first n terms (S_n) of an AP: S_n = n/2 (a + l)
l is the last term, and the formula calculates the sum by averaging the first and last terms. Useful when the last term is known.
Common difference (d): d = a_(k+1) - a_k
d is the common difference; a_(k+1) and a_k are successive terms in the AP. Helps in verifying if a sequence is an AP.
n-th term from the last (a_m) of an AP: a_m = l - (m - 1)d
a_m is the m-th term from the last, l is the last term, and d is the common difference. This is useful when counting backwards from the last term.
Total number of terms (n): n = (l - a)/d + 1
n gives the total count of terms between the first term (a) and the last term (l) with a common difference (d). Ideal for determining the number of elements in an AP.
If (a, d) are given, the first four terms are: a, a+d, a+2d, a+3d
This construction derives the first four terms directly using the first term (a) and common difference (d). Simplifies the process of generating terms.
Infinite AP: a, a+d, a+2d, ...
In an infinite AP, terms continue indefinitely. Understanding its structure helps in identifying unbounded sequences.
Finite AP: a, a+d, a+2d, ... , l
A finite AP has a last term (l). This concept is essential in distinguishing between bounded and unbounded series.
Identifying an AP: Check if d = a_(k+1) - a_k is constant.
This method verifies if a sequence forms an AP by checking the consistency of the difference across terms.
Equations
3, 8, 13, 18, ...
This sequence forms an AP with a common difference of 5 (d = 8 - 3 = 5). Identifies a numeric example of an AP.
10, 7, 4, 1, ...
This sequence is an AP where d = -3 (4 - 7 = -3). Highlights how negative differences work in an AP.
-2, 0, 2, 4, ...
An AP example with d = 2. It illustrates how sequences can cross zero.
1, 3, 5, 7, ...
AP with d = 2, showing a consistent pattern of odd numbers. A common example in numeric discussions.
5, 10, 15, 20, ...
This series represents an AP with d = 5, commonly used to represent increments in real-life scenarios such as savings.
4, 4, 4, 4, ...
An AP where d = 0, showing how a constant value remains unchanged over multiple terms.
11, 8, 5, 2, ...
An AP with d = -3, useful to demonstrate decreasing sequences.
-1, -2, -3, -4, ...
An AP where d = -1, applying to scenarios that involve steady reduction.
0, 1, 2, 3, ...
A classic AP with d = 1, commonly used to demonstrate counting.
12, 10, 8, 6, ...
An AP illustrating decreasing numbers, with d = -2. Good for examples of limited resources decreasing over time.
