Question

数列5,45,445,4445,…,444…5的一个通项公式an=?

Answer 1

A2 - A1 = 45 - 5 = 40 = 4 * 10^1
A3 - A2 = 445 - 45 = 400 = 4 * 10^2
A4 - A3 = 4445 - 445 = 4000 = 4 * 10^3
……
An - An-1 = …… = …… = 4 * 10^(n-1)
以上各式连加可得
An - A1 = 4 * (10^1 + 10^2 + …… + 10^(n-1)) = 4 * Tn
Tn = 10^1 + 10^2 + …… + 10^(n-1)为以10为首项,10为公比的等比数列前n-1项的和
Tn = 10 * (10^(n-1) - 1) / (10 - 1) = (10^n - 10) / 9
An = 4*Tn + 5
= (4/9) * (10^n - 10) + 5
以上An即为所求数列的通项公式.