Defined by geometric series?

a2=a 1*q？

a3=a2*q？

a(n- 1)=a(n-2)*q？

An = a (n- 1) * q * * n- 1。 What is the sum of the two sides of the equation?

a2+a3+...+an=[a 1+a2+...+a(n- 1)]*q？

That is Sn-a 1=(Sn-an)*q, that is, (1-q)Sn=a 1-an*q?

When q≠ 1, sn = (a1-an * q)/(1-q) (n ≥ 2)?

It also holds when n= 1

When q= 1, Sn=n*a 1?

So sn = n * a1(q =1); (a 1-an * q)/( 1-q)(q≠ 1)。

2. Derivation of summation formula of equal proportion sequence

Dislocation subtraction

Sn=a 1+a2 +a3 +...+ An

Sn *q = a1* q+a2 * q+...+a (n-1) * q+an * q = a2+a3+...+an+an * q

The above two expressions are subtracted to get (1-q) * sn = a1-an * q.

3. Derivation of summation formula of equal proportion sequence

complete induction

It is proved that (1) when n= 1, the left =a 1, and the right = A 1 Q0 = A 1, the equation holds;

(2) Suppose that when n=k(k≥ 1, k∈N*), the equation holds, that is, AK = a1qk-1;

When n=k+ 1, AK+1= akq = a1qk = a1q (k+1)-1;

That is to say, when n=k+ 1, the equation also holds;

Judging from (1)(2), the equation holds for all n∈N*.

References:

Baidu Encyclopedia Entry-Sum Formula of Equal Proportional Sequence