As An=BnCn, where {Bn} is arithmetic progression and {Cn} is geometric progression; List Sn separately, and then multiply all formulas by the common ratio q of geometric series at the same time, that is, q Sn；; Then stagger one bit and subtract the two expressions. This method of summation of series is called dislocation subtraction.
Typical example: sum sn =1+3x+5x2+7x3+…+(2n-1) xn-1(x ≠ 0, n∈N*).
When x= 1, sn =1+3+5+…+(2n-1) = N2.
When x≠ 1, sn =1+3x+5x2+7x3+…+(2n-1) xn-1.
∴xsn=x+3x2+5x3+7x4+…+(2n- 1)xn
The two expressions are subtracted to get (1-x) sn =1+2 (x+x2+x3+x4+…+xn-1) xn.
Extended data:
A series in which the difference between each term and its previous term is equal to the same constant, usually expressed by A and P. This constant is called arithmetic progression's tolerance and is often expressed by the letter D..
For example: 1, 3, 5, 7, 9...2n- 1. The general formula is: an = a1+(n-1) * D. The first term a 1= 1, and the tolerance d=2. The first n terms and formulas are: sn = a1* n+[n * (n-1) * d]/2 or Sn=[n*(a 1+an)]/2. Note: All the above n are positive integers.
A geometric series with all positive numbers, taking the same base, constitutes a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.
If (an) is a geometric series, all terms are positive, and the common ratio is q, then (the logarithm of an based on log) is arithmetic, and the tolerance is the logarithm of q based on log. The sum of the first n terms in geometric progression is sn = a1(1-q n)/(1-q) = a1(q n-1)/(q-1) = (a In geometric series, the first term A 1 and the common ratio q are not zero.
Baidu Encyclopedia-arithmetic progression
Baidu encyclopedia-geometric series