求有限项级数 1×2×2^n+2×3×2^(n-1)+3×4×2^(n-2)+…+(n+1)×(n+2)×2^0 之和

myyour

级数求和：
1×2×2^n+2×3×2^(n-1)+3×4×2^(n-2)+…n×(n+1)×2^1+(n+1)×(n+2)×2^0

王守恩

    LinearRecurrence[{5, -9, 7, -2}, {0, 0, 0, 2}, 40]
{0, 2, 10, 32, 84, 198, 438, 932, 1936, 3962, 8034, 16200, 32556, 65294,
130798, 261836, 523944, 1048194, 2096730, 4193840, 8388100, 16776662,
33553830,67108212,134217024,268434698,536870098,1073740952,2147482716,
4294966302, 8589933534, 17179868060, 34359737176, 68719475474, 137438952138,
274877905536, 549755812404, 1099511626214,......}

    \(5×002-9×00+7×00-2×00=10\)
    \(5×010-9×02+7×00-2×00=32\)
    \(5×032-9×10+7×02-2×00=84\)
    \(5×084-9×32+7×10-2×02=198\)
    \(5×198-9×84+7×32-2×10=438\)

     或：\(a(n)=2^n+n-n^2-2\)

myyour

王守恩 发表于 2021-4-20 08:36
LinearRecurrence[{5, -9, 7, -2}, {0, 0, 0, 2}, 40]
{0, 2, 10, 32, 84, 198, 438, 932, 1936, 3962, 8 ...

王老师我看不懂啊。

myyour

突然发现把上面的n换成n+4，得Sn=2^(n+4)-n^2-7n-14，刚刚好。

luyuanhong