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Formulas for Arithmetic Progression
Formulas To Solve Arithmetic Progression in Aptitude
An arithmetic progression or arithmetic sequence is a of numbers such that the difference of any two successive members is a constant.Where common difference is denoted by d.n-th term of an arithmetic progression denoted by an Sum of the first n elements denoted by Sn
Basic Concept on Arithmetic Progression
| a, a + d, a + 2d, a + 3d ……… |
| a |
| d |
Example: Given sequence or series is 2, 4, 6, 8, 10,…
Here, a = 2 and d = 2
Finite or Infinite Arithmetic Progressions
- Finite Arithmetic Progression
When there are limited number of terms in the sequence then it is known as Finite Arithmetic Progression.
For example: 10, 20, 30, 40, 50
- Infinite Arithmetic Progression
When there are unlimited number of terms in the sequence then it is known as Infinite Arithmetic Progression.
For example: 3, 5, 7, 9, 11, 13, ..…….
Formula of Arithmetic Progression
- nth term of an AP
Formula to find the nth term of an AP is
Tn = a + (n – 1) d
where tn = nth term,
a= the first term ,
d= common difference,
n = number of terms in the sequence.
Number of terms in an AP
- Formula to find the numbers of term of an AP is
n = \left [ \frac{(l-a)}{d} \right ] + 1
where
n = number of terms,
a= the first term,
l = last term,
d= common difference.
Sum of first n terms in an AP
- Formula to find the tsum of first n terms of an AP is
S_{n} = \frac{n}{2} [2a + (n-1)d]
OR
S_{n} = \frac{n}{2} (a+l)
where,
a = the first term,
d= common difference,
l = tn = nth term = a + (n-1)d
Arithmetic Mean
If a, b, c are in AP, then the Arithmetic mean a and c is b i.e.
b = \frac{1}{2} (a + c)
Some other important formulas of Arithmetic Progression
- Sum of first n natural numbers
We derive the formula to find the sum of first n natural numbers
S = \frac{n (n+1)}{2}
where
S = Sum of first n natural numbers
n = number of natural numbers
Sum of squares of first n natural numbers
- Formula to find the sum of squares of first n natural numbers of an AP is
S= \frac{{ n (n+1) (2n+1) }} {6}
where
S = Sum of first n natural numbers
n = number of natural numbers.
Sum of first n odd numbers
- Formula to find the nth term of an AP is the square of the number of terms
S =n2
where
S = Sum of first n natural numbers
n = number of natural numbers
Sum of first n even numbers
- Formula to find the sum of of an AP is
S = n(n+1)
where
S = Sum of first n natural numbers
n = number of natural numbers
Properties of Arithmetic Progression
- If a fixed number is added or subtracted from each term of an AP, then the resulting sequence is also an AP and it has the same common difference as that of the original AP.
- If each term in an AP is divided or multiply with a constant non-zero number, then the resulting sequence is also in an AP.
- If nth is in linear expression then the sequence is in AP.
- If a1, a2, a3, …, anand b1, b2, b3, …, bn, are in AP. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in AP.
- If nth term of a series is Tn = An + B, then the series is in AP
- Three terms of the A.P whose sum or product is given should be assumed as a-d, a, a+d.
- Four terms of the A.P. whose sum or product is given should be assumed as a-3d, a-d, a+d, a+3d.
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Nice content
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