{an}是首项为a1=1,公差为d的等差数列,求3^a1+3^a2+3^a3+……+3^an
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![{an}是首项为a1=1,公差为d的等差数列,求3^a1+3^a2+3^a3+……+3^an](/uploads/image/z/2725321-49-1.jpg?t=%EF%BD%9Ban%EF%BD%9D%E6%98%AF%E9%A6%96%E9%A1%B9%E4%B8%BAa1%3D1%2C%E5%85%AC%E5%B7%AE%E4%B8%BAd%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%B1%823%5Ea1%2B3%5Ea2%2B3%5Ea3%2B%E2%80%A6%E2%80%A6%2B3%5Ean)
{an}是首项为a1=1,公差为d的等差数列,求3^a1+3^a2+3^a3+……+3^an
{an}是首项为a1=1,公差为d的等差数列,求3^a1+3^a2+3^a3+……+3^an
{an}是首项为a1=1,公差为d的等差数列,求3^a1+3^a2+3^a3+……+3^an
依照题意,a1=1.a2=1+d,a3=1+2d,...,an=1+(n-1)d
则3^a1+3^a2+3^a3+……+3^an
=3^1+3^(1+d)+3^(1+2d)+……+3^[1+(n-1)d]
=3*[(3^d)^0+(3^d)^1+(3^d)^2+...+(3^d)^(n-1)] 设3^d=A
=3*[A^0+A^1+A^2+...+A^(n-1)]
=3*(1-A^n)/(1-A) 再将A=3^d代入
=3*(1-3^nd)/(1-3^d)
3^n+(n(n-1)d/2)