✨✔️♦️derivation of formula in Ap is important ???? according to exm point of u .... [CBSE class 10th ]
Q: if yes then please derive the formula
Answers
Answer:
I think derivation of AP has not been asked in any of the examinations.....
So.... Don't prepare AP derivation.... It won't come.... Most probably...
Thank you
Jalay
Answer:
Explanation:
d = common difference
a1 = first term
a2 = second term
a3 = third term
am = mth term or any term before an
an = nth term or last term
d=a2−a1=a3−a2=a4−a3 and so on.
then,
Derivation for an in terms of a1 and d
a1=a1
a2=a1+d
a3=a2+d=(a1+d)+d=a1+2d
a4=a3+d=(a1+2d)+d=a1+3d
a5=a4+d=(a1+3d)+d=a1+4d
...
am=a1+(m−1)d
...
an=a1+(n−1)d
similarly,
an=an
an−1=an−d
an−2=an−1−d=(an−d)−d=an−2d
an−3=an−2−d=(an−2d)−d=an−3d
an−4=an−3−d=(an−3d)−d=an−4d
...
am=an−(n−m)d
...
a1=an−(n−1)d
Derivation for the Sum of Arithmetic Progression, S n
Sn=a1+a2+a3+a4+...+an
Sn=a1+(a1+d)+(a1+2d)+(a1+3d)+...+[a1+(n−1)d] ← Eq. (1)
Sn=an+an−1+an−2+an−3+...+a1
Sn=an+(an−d)+(an−2d)+(an−3d)+...+[an−(n−1)d] ← Eq. (2)
Add Equations (1) and (2)
2Sn=(a1+an)+(a1+an)+(a1+an)+(a1+an)+...+(a1+an)
2Sn=n(a1+an)
Sn=n2(a1+an)
Substitute an = a1 + (n - 1)d to the above equation, we have
Sn=n/2{a1+[a1+(n−1)d]}
Sn=n/2[2a1+(n−1)d]
THIS IS NOT FOR EXAM POINT OF VIEW...
but i did it for helping