Question

Find the sum of 1+3+5+7+..............+99​Step by step​

Answer 1

Answer:

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Answer 2

Question :Find the sum of 1+3+5+7+..............+99

Answer :

The above series forms an Arithemetic Progression (A.P).

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.

For example,

the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1.

Second example: the sequence 3, 5, 7, 9, 11,... is an arithmetic progression

with common difference 2.

Now the above series

1+3+5+………+97+99

We can solve this in two ways

Solution:

Using Progression Concept

The Common Difference is 3–1 = 2

Total no of terms can be found using below formula

T1 = a+(n-1)d

Tn=a+(n−1)d

Tn is the nth Term

a is the First Ter

n is the no of Terms

d is the common difference

Here Tn is 99, d=2

99 = 1+(n-1)2;

98= 2n-2;

2n = 100;

n = 50;

There are total 50 terms in this series. Sun can be found using below formula

Sum S=n/2a+l

Where n is the no of terms

a is the first term and l is the last term

S = 50/2{1+99}

S = 25*100;

S = 2500

You can skip step where no of terms are calculated as it is known first 100 terms will have 50 even and 50 odd numbers, Clearly the above series is sum of odd numbers less than 100, there are total 50 odd and 50 even in first 100 natural numbers. So n = 50.

Step-by-step explanation:

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