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there are many in progression who to know these pattern.can u give me all the pattern that i will come across.
If the first term of the Arithmetic Progression which is generally called as A.P. is ‘a’ and the common difference is ‘d’ then the nth term is given by tn = a+(n-1)d.A sequence of numbers <an> is said to be in arithmetic progression if the difference between the terms is constant i.e. tn – tn-1 is a constant for all n ∈ N. the constant difference between the terms is called the common difference and is denoted by‘d’.
Similarly, the sum of first ‘n’ terms of the A.P. is given by
Sn= n/2 [2a + (n-1) d]
= n/2 [a + l], where l is the last term of the A.P.
If the first term of the geometricprogression is ‘a’ and the common difference is ‘r’ then the nth term is given by tn = arn-1If every succesiveterm is a fixed multiple of the preceding term then the series is said to be a geometric series. A progression of the form a, ar, ar2, ….. is a geometric progression where the common multiple r is called the fixed ratio. it follows from the definition itself that no term of the G.P can be zero because in such a case the series will get reducedto a zero series.
The sum Snof the first ‘n’ terms of the GP is given by
Sn= a(rn-1)/ (r-1) , if r ≠1
= na, if r =1.
The nth term of H.P. is given by the sequence a1, a2, …. an, where ai ≠ 0 for every i is said to be in harmonicprogression (H.P.) if the sequence 1/a1, 1/a2, …. ,1/an is in A.P.
an=1 / [a+(n-1)d], where a = 1/a1and d = 1/a2-1/a1.
limn→∞Sn= ab/1–r + dbr/(1–r)2.
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