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What is cumulative multiplication? How to use it?
Accumulation is to make use of the fact that the front and back terms (generally 2 or 3 terms) have the same terms and opposite coefficients. For example, an = 1 / n-1 / (n + 1)..., and an-1 = 1 / (n-1) - 1 / N have the same term 1 / n and opposite coefficients,
So Sn = a1 + A2 + a3 +. + an = 1-1 / 2 + 1 / 2-1 / 3 + 1 / 3-1 / 4 +. + 1 / (n-1) - 1 / N + 1 / n-1 / (n-1) = 1-1 / (n-1)
Another example
BN = 1 / n-1 / (n + 2), then bn-1 = 1 / (n-1) - 1 / (n + 1), BN-2 = 1 / (n-2) - 1 / N, then BN and BN-2 have the same term 1 / N, the coefficient is opposite. Sn = 1-1 / 3 + 1 / 2-1 / 4 + 1 / 3-1 / 5 +. + 1 / (n-2) - 1 / N + 1 / (n-1) - 1 / (n + 1) + 1 / n-1 / (n + 2) = 1 + 1 / 2-1 / (n + 1) - 1 / (n + 2)
Multiplication is similar
bn=n/(n+1),bn-1=（n-1)/n,
cn=b1*b2*b3*b4*.*bn-1*bn=1/2*2/3*3/4*.(n-1)/n*n/(n+1)=1/(n+1)...
bn=n/(n+2),bn-1=(n-1)/(n+1),bn-2=(n-2)/n,
cn=b1*b2*b3*.bn-2*bn-1*bn=1/3*2/4*3/5*4/6*.*(n-2)/n*(n-1)/(n+1)*n/(n+2)
=1*2/((n+1)*(n+2))
Under what circumstances is the cumulative addition used and under what circumstances is the cumulative multiplication used? Are these two methods used to find the sum of sequences?
If the latter term is added with the former term, it can reduce part of the total, and if the latter term is multiplied with the former term, it can reduce part of the total. Generally speaking, cumulative addition can be used to deduce the general term formula and sum, while cumulative multiplication is only used to deduce the general term formula
Urgent: sum: SN = - 1-5 / 9-7 / 27 +. + (- 2n-1) (1 / 3) ^ n thank you for 30 points
Sum (?) (?) / Sn = - 7 + / sn-1)... - 1
If you want to be right, add 30
-Sn = 1 + 5 / 9 + 7 / 27 +... + (2n + 1) / 3 ^ n-sn = 1 + (2 * 2 + 1) / 3 ^ 2 + (2 * 3 + 1) / 3 ^ 3 +... + (2n + 1) / 3 ^ n -- 1-sn / 3 = 1 / 3 + (2 * 2 + 1) / 3 ^ 3 + (2 * 3 + 1) / 3 ^ 4 +... + (2n-1) / 3 ^ n + (2n + 1) / 3 ^ n + (n + 1) - 21-2 get - 2 / 3Sn = 2 / 3 + (2 * 2 + 1) / 3 ^ 2 + 2 / 3 ^ 4 + 2 / 3 ^ 5 +... + 2 / 3 ^ n
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