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By now you wouldโve started planning your masters abroad, and you know that finding the funds to get to campus isnโt always easy. Prodigy Finance is here to change that
I have posted a video on YouTube to discuss Arithmetic and Geometric Progression
Attached pdf of this Article as SPOILER at the top! Happy learning!
Following is Covered in the Video
Theory
What is Arithmetic Progression (AP)? AP Formulas AP Problems What is Geometric Progression (GP)? GP Formulas GP Problems Miscellaneous Problems
What is Arithmetic Progression (AP)?
A sequence of numbers such that the difference between the consecutive terms is constant. It is also known as Arithmetic Sequence or Arithmetic Series Example: 2 , 5 , 8 , 11โฆ. ( Consecutive terms have the same common difference of 3 )
AP Formulas
\(N^{th}\) Term of an Arithmetic Series
Arithmetic Series is given by a , a+d, a+2d,... \(T_{1}\) = a = a + (1-1)d \(T_{2}\) = a + d = a + (2-1)d \(T_{3}\) = a + 2d = a + (3-1)d . . . \(T_{n}\) = a + (n-1)d
\(N^{th}\) of an Arithmetic Series, \(T_{n}\) = a + (n-1)d where, a is the first term of the sequence d is the difference between consecutive terms in the sequence (common difference) n is the number of terms \(T_{n}\) is the nth term in the sequence
\(S_{n}\) = Number of terms * Mean of First term and Last term
Number of terms in an AP is given by
n = \(\frac{(T_{n} โ T_{1}) }{๐ }+ 1\)
For Arithmetic Series Mean = Median = Avg. of 1st and Last term = Avg. of 2nd term from the starting and second term from the end = Avg. of 3rd term from the starting and third term from the end and so on....
AP Problems
Q1. Find the number of terms in the series 3,4,5,โฆ,21
Sol: Number of terms = 19, Check Video for solution
Q2. Find the sum of first โnโ positive integers (i.e. 1 + 2 + 3 +โฆ + n)
Sol: Series is given by 1, 2, 3, 4, โฆ, n
Sum of the series = Number of terms * Mean of First and Last term = n * \( \frac{ ((1+๐))}{2}\)
Sum of first n positive integers = \(\frac{(๐โ(๐+๐))}{๐}\)
Sum of First n Positive integers
Sum of first n integers = \(\frac{(๐(๐+๐))}{๐}\)
This can be used only when
Terms are starting from 1 and Series comprises of consecutive integers
Q6. If the first term of a sequence is 2, the last term of the sequence is 44 and the number of terms is 15. Find the sum of all the terms of the sequence?
Geometric Series is a series in which consecutive terms have the same ratio. It is also known as Geometric Sequence or Geometric Series Example: 2 , 6 , 18 , 54โฆ. ( Consecutive terms have the same ratio of 3:1 )