Arithmetic Progression: Sum of n terms

So, I'm continuing in this post with AP.

    I hope you've gone through our previous post which was focused on arithmetic sequence and Basics,

• If not, click here for reading that post.

 • So, I've revealed the formula of last term of an arithmetic sequence in that post, Now here I'm going to share the formula of finding the sum of n terms of an arithmetic sequence.

Sum of "n" terms of an Arithmetic sequence

  •  Finding Sum of given terms of an AP:-
let there are n terms in given progression and the sum of those terms are defined​ as "S" and formulated as,

S = n/2 [2a + (n-1)d]

all other values are known i.e.
"a" as first term and
"d" as common difference.

  • This can formula can also be utilised for evaluating other unknown values if sum is given. That means in this there four (S, a, n, d) variables and any one can be calculated if other three are given.

• Another form of sequence formula

yes, this formula has an another formation too. 

If we have last term of AP i.e. (Tn or l) then formula can be shortened like

S = n/2 [ a+ Tn]

  •  again, this formula can be modified for finding sum of "n" consecutive numbers:

 S= [ n (n+1)] / 2

this formula can be derived as below:

we have 

S = n/2 [ 2a + (n-1)d]

for n consecutive number 

a= 1,  d= 1

so,

S = n/2 [ 2×1 + (n-1)×1]

   = n/2 [ 2  + n-1]

   = n/2 [ n+1 ] 

 hence,

S = [n( n+ 1 )] / 2

you can find an another way of derivation of this sequence formula click here for derivation which is quite easy and it is fun to learn in that method, I personally recommend this post read this at least once.

And also comment about this tutorial how do you feel about this and what else do you want to add here.

Arithmetic Progression (sequence)

Hey learners,
hope you are doing well.

Today I'm here with Arithmetic progression topic, for this topic I can say it has given an access to hidden treasure of Mathematics .

So, I'm starting with general intro with some basics of arithmetic progression .

What is Arithmetic progression (AP) /Arithmetic Sequence

• Definition:

progression are refers to series formations,

and those series formations in which the terms are lying at same gaps are called as Arithmetic progression or AP.

further more I can state that in this series the difference between two consecutive terms remain same Examples:

2, 4, 6, 8......

1, 3, 5, 7......

• How long the series can be ?

This doesn't have any criteria or any boundation, means the number of terms of series can be infinite, unless number of terms given in the question.

• What Significance arithmetic series has ?

A number of significance this topic contains among those some are, (with proper values and formula we can evaluate)

•  We can identify the number of terms of any series
•  We can identify the missing terms.
•  We can identify the value of last terms.
• We can find the sum of all the terms of given series

let we are having any series 

• 5, 10, 15, 20, 25, 30.......up to 35th term

now it is being crucial to add one after one to find the sum of all the terms like

 5 + 10 + 15 +20 +.......up to 35th term.

(nobody wants to do this for each problem)

to overcome from this, AP and its applications are derived.

 •  similarly we can also evaluate 35th term directly.

 • I think I've given complete clarification regarding Arithmetic progression and it's existence.

 • In this post I'm revealing the Formula of AP only, further discussion on topic will be carry forwarded to our next posts.

Formula of AP:

         T = A + (n-1)D

where,

T - last term, nth term of series.

A - first term of series.

n - number of terms.

D - common difference between to consecutive terms.

so,

If you still have some doubts, let me know in comment section, I'll be pleased to hear from you.

Stay tuned for the same topic for formula and Q/SOL. and many more...

see you in next post.

Thanks.