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Explanation:
What number would come next in the series 7, 9, 13, 21, 37?
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175 Answers

Sarvesh Kumar
, Test Engineer at Infosys Limited (2020-present)
Answered 4 years ago
According to mathematics “set of related terms to each other is known as sequence and sum of these terms(expression of sum) indicates series.”
Here is the problem 7,9,13,21,37,?
Every person has various ways to determine the solution, I have also a solution i.e.
Let us assume that sum of all the terms(which are know as well as unknown) is Sn.
Sn=7+9+13+21+37+………+A(n-1)+An.____say eq.1
Now if we take this sum in such a way …….
Sn=_7+9+13+21+37+……..+A(n-2)+A(n-1)+An.
Say it eq.2
Now subtract both the equations.
eq.1-eq.2
Sn=7+9+13+21+37+…..A(n-1)+An.
Sn=_+7+9+13+21+……A(n-2)+A(n-1)+An.
After subtracting we get
0=7+[2+4+8+16+32+………A(n-1)]-An
Replacing An in RHS.
An=7+[2+4+8+16+32+…….A(n-1)]______eq(*)
Here terms inside big bracket is known as geometrical progression( GP)
Geometrical progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Sum of GP=A+Ar^1+Ar^2+…Ar^n.
So the sum of GP, first term is A and common ratio is r having (n-1) terms….. is {A*[r^(n-1)-1]}/{[r-1]}
Calculation of r(common ratio) is [second term/first term] or [third term/second term] or [An/A(n-1)].
So that from eq(*)
An=7+{2*[2^(n-1)-1]}/{[2–1]}
An=7+(2^n-2)/1
An=7+2^n-2
An=5+2^n…(this is the general term)
Put values of n=1,2,3,4,5,6……..n
A1=5+2^1=7
A2=5+2^2=9
A3=5+2^3=13
.
.
A6=5+2^6=69(this is the answer)
So sequence will be …7,9,13,21,37, 69,133….
Thanks …!!!
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Darren Mistry
, R&D Engineer at Samsung Research Institute Noida (2019-present)
Answered 4 years ago · Author has 139 answers and 348.9K answer views
The trick for solving these kind of problems is to first check for geometric ratio or the common difference between every two terms. Here when we look for geometric ratio we don't find anything promising. No worries, we check for the common difference!
The following observations one is supposed to make:
a1−a0=2=21a1−a0=2=21
a2−a1=4=22a2−a1=4=22
a3−a2=8=23a3−a2=8=23
and so on…
Now once you can find the pattern, always try for the general term of the sequence. Here in this case the recursive definition will be,
a0=7a0=7
an=an−1+2nan=an−1+2n ,n>