Explain arithmetic progressions??
Answers
Introduction, basic concept:-
Arithmetic Progressions (Abbreviated as A.P) is any series whose common difference is the same in all cases.
You might be confused by common difference, let me explain.
Suppose you have two random series:-
a)1,2,3,4,5,6
And
b)2,4,9,16,25,36,49
Now, which one of these series do you think is an Arithmetic Progression?
Let's verify and check it out.
Common difference (d) is basically the difference between two terms. It can be written mathematically as:-
d1=t1-t2; d2=t3-t2; d3=t4-t3 and so on till n terms.
If in the above set of formulae, d1, d2, d3,...dn are all equal, then the series is said to be in an Arithmetic Progression.
Let us take the two examples a and b mentioned before,
a)1,2,3,4,5,6
Now, let us see the common differences,
d1=t2-t1=2-1=1
d2=t3-t2=3-2=1
d3=t4-t3=4-3=1
d4=t5-t4=5-4=1
d5=t6-t5=6-5=1
d1=d2=d3=d4=d5
Hence, the series is in arithmetic progression.
b)2,4,9,16,25,36,49
Let us do the same as we did in a)
d1=t2-t1=4-2=2
d2=t3-t2=9-4=5
Here d1≠d2 and hence this series is not in arithmetic progression.
And that's it. This is the very basic concept of Arithmetic Progression.
Order of an AP:-
It is too simple, no rocket science, though...
It is given by a, (a+d), (a+2d), (a+3d),...
Where a is the first term and d is the common difference.
You can verify it for any given AP, it always is the same...
Nth term of an AP:-
Suppose you have an AP which goes:-
2,4,6,8,10,12,14,...
And you are asked to find out the 100th term of the AP,
Will you rather keep adding the common difference (which is 2) 100 times from the start?
Of course not, that would be tedious. And hence there is a formula which would help find that term easily.
And that formula is very simple, it is:-
Tn=a+(n-1)d
Where Tn is the nth term, a is the first term of the AP, n the number of the term (say, 100th term as in the example) and d, as you know, is the common difference.
Using this formula, you can find any term of the given AP very easily.
Let us take that earlier example and find the 100th term,
Tn=a+(n-1)d
T100=2+(100-1)2
T100=2+99×2
T100=2+198
T100=200
Hence, the 100th term of that AP is 200.
Sum of n terms of an AP:-
Let us take the same previous example,
2,4,6,8,10,12,...
And if you are asked to find the sum of 30 terms, don't worry, another simple formula is there to help...
And it is:-
Sn=n/2 × [2a+(n-1)d]
Where Sn is the sum of n terms, a is the first term, n is the nth term, d is the common difference.
Let us use that example.
Sn=n/2×[2a+(n-1)d]
S30=30/2×[2(2)+(30-1)(2)]
S30=15×[4+58]
S30=15×62
S30=930
Hence the sum of 30 terms of that AP is 930.
Conclusion:-
I've explained all the basics necessary for you to understand when it comes to an AP
You can solve sums using these explanations and get to know more about the topic and understand in great detail.
Thanks for reading. Hope it helps. Lots of love