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Hi All,
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 SeriesArithmetic 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
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.
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\(T_{n}\) = a + (n-1)d
\(N^{th}\) of an Arithmetic Series,
\(T_{n}\) = a + (n-1)dwhere,
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
Sum of n terms of a AP is given by\(S_{n} = \frac{๐ง}{๐}[2a + (n-1)d] \) \(S_{n} = n*\frac{(๐+(๐+(๐งโ๐)๐))}{๐ }\)\(S_{n}\) = Number of terms * Mean of First term and Last termNumber of terms in an AP is given byn = \(\frac{(T_{n} โ T_{1}) }{๐ }+ 1\)
For Arithmetic SeriesMean = 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 ProblemsQ1. Find the number of terms in the series 3,4,5,โฆ,21Sol: 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 integersSum 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
Q3. Find the sum of first 50 positive integers.Sol. 1275. Check
Video for solution
Q4. Find the sum of all the integers between 40 and 100 inclusive.Sol. 4270. Check
Video for solution
Q5. Which term of the sequence 1,4,7,10,โฆ is 43?Sol. 15. Check
Video for solution
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? Sol. 345. Check
Video for solution
What is Geometric Progression (GP)?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 )
GP Formulas\(N^{th}\) Term of a Geometric SeriesGeometric Series is given by
\(a , ar, ar^2, ar^3, โฆ , ar^{n-1}\)
\(T_{1}\) = a = \(ar^{1-1}\)
\(T_{2}\) = ar = \(ar^{2-1}\)
\(T_{3}\) = a\(r^2\) = \(ar^{3-1}\)
.
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\(T_{n}\) = \(ar^{n-1}\)
Sum of n terms of a GP is given by\(S_{n} = a*\frac{((๐^๐ โ ๐)}{(๐ซโ๐))}\)GP ProblemsQ1. Find the sum of first 10 terms of a Geometric series whose first term is 3 and common ratio is 2.Sol. 3069. Check
Video for solution
Q2. Which term of the geometric series 4,8,16,โฆ is 4096?Sol. 11. Check
Video for solution
Miscellaneous ProblemsFollowing is not an Arithmetic or a Geometric series. Find the \(n^{th}\) term of this series \(T_{n}\) of the series:1. 1, 4, 9, 16,โฆ
2. 1, 8, 27, 64, โฆ
3. 2, 5, 10, 17,โฆ
4. 2, 9, 28, 65,โฆ
5. 2, 6, 12, 20,โฆ
6. 2, 4, 8, 16, โฆ
7. 1, 2, 3, 4, 5, 8, 7, 16,โฆ
8. 1, 2, 3, 5, 8, 13,โฆ
9. 1, 2, 2, 4, 8, 32, โฆSol. Check
Video for solution
1. \(T_{n}\) = \(n^2\)
2. \(T_{n}\) = \(n^3\)
3. \(T_{n}\) = \(n^2\) + 1
4. \(T_{n}\) = \(n^3\) + 1
5. \(T_{n}\) = \(n^2\) + n
6. \(T_{n}\) = 2n
7. \(T_{Odd}\) = n \(T_{Even}\) = \(2^{n/2}\)
8. \(T_{n}\) = \(T_{n-1}\) + \(T_{n-2}\) for n>= 3
9. \(T_{n}\) = \(T_{n-1}\)* \(T_{n-2}\) for n>= 3
Hope it Helps!
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