![Sum of nth term
Sum = n x [(first term + last term)/ 2]
Now last term will be a + (n-1)d.
,’, Sum,S= n x [{a + a + (n-1)d} / 2]
= n/2 [2a + (n-1)d]](https://image.slidesharecdn.com/arithmeticprogressions-151112080002-lva1-app6891/85/Arithmetic-progressions-11-320.jpg)
![Derivation
The sum to n terms is given by:
Sn = a + (a+d) + (a+2d) + .... + [a + (n-1)d (1)
If we write this out backwards, we get:
Sn = [a + (n-1)d] + [a + (n-2)d] +... + a (2)
Now let’s add (1) and (2)
2Sn = [2a + (n-1)d] + [2a + (n-1)d] ........... +
[2a + (n-1)d]
So, Sn = n/2 [2a +(n-1)d]](https://image.slidesharecdn.com/arithmeticprogressions-151112080002-lva1-app6891/85/Arithmetic-progressions-12-320.jpg)
![The difference between two
terms of an AP can be
formulated as below:-
nth term – kth term
= t(n) – t(k)
= [a + (n-1)d] + [a + (k-1)d]
= a + nd – d - a + kd + d
= nd + kd
Hence, t(n) – t(k) = (n + k)d](https://image.slidesharecdn.com/arithmeticprogressions-151112080002-lva1-app6891/85/Arithmetic-progressions-13-320.jpg)
Arithmetic progressions
- 2. What is arithmetic progressions? • Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.
- 3. What is sequence? • Sequence : A list of numbers having specific relation between the consecutive terms is generally called a sequence.
- 4. Examples • For instance, the sequence 5, 7, 9, 11, 13, 15,...... is an Arithmetic Progression.
- 5. Illustrative examples a = d, where d =1 a+2da+d a+3d
- 6. • The general form of A.P. is a, a+d, a+2d, a+3d……………, a +(n-1)d where a is the first term and d is called common difference.
- 7. What is a common difference? • Common Difference: The fixed number which is obtained by subtracting any term of AP from its previous term. • It is denoted as d.
- 8. • If we take First term of an AP as a and common difference as d, then nth term of an AP will be An= a + (n-1)d
- 9. • 3, 7, 11, 15, 19............ d= 4, a=3 Notice in this sequence that if we find the difference between any term and and the term before it we always get 4. 4 is then called common difference and is denoted with the letter d. To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: an = an-1 + 4 The first term in the sequence would be a1 which is sometimes just written as a.
- 10. 3, 7, 11, 15, 19........ d= 4, a=3 an = a + (n-1)d Try this to get the 5th term. a5 = 3 + (5-1)4 = 3 + 16 = 19 +4 +4 +4 +4
- 11. Sum of nth term Sum = n x [(first term + last term)/ 2] Now last term will be a + (n-1)d. ,’, Sum,S= n x [{a + a + (n-1)d} / 2] = n/2 [2a + (n-1)d]
- 12. Derivation The sum to n terms is given by: Sn = a + (a+d) + (a+2d) + .... + [a + (n-1)d (1) If we write this out backwards, we get: Sn = [a + (n-1)d] + [a + (n-2)d] +... + a (2) Now let’s add (1) and (2) 2Sn = [2a + (n-1)d] + [2a + (n-1)d] ........... + [2a + (n-1)d] So, Sn = n/2 [2a +(n-1)d]
- 13. The difference between two terms of an AP can be formulated as below:- nth term – kth term = t(n) – t(k) = [a + (n-1)d] + [a + (k-1)d] = a + nd – d - a + kd + d = nd + kd Hence, t(n) – t(k) = (n + k)d
- 14. Thank You!
- 15. Name:Shefali Kumar Class:X-B Roll No: 28 Topic: Arithmetic Progressions School: Ramagya School, Noida, Sector- 50, Uttar Pradesh