数列an的通项公式an=6n-5(n为奇数),an=2的n次方(n为偶数),求数列an的前n项的和Sn
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![数列an的通项公式an=6n-5(n为奇数),an=2的n次方(n为偶数),求数列an的前n项的和Sn](/uploads/image/z/200186-26-6.jpg?t=%E6%95%B0%E5%88%97an%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8Fan%3D6n-5%EF%BC%88n%E4%B8%BA%E5%A5%87%E6%95%B0%EF%BC%89%2Can%3D2%E7%9A%84n%E6%AC%A1%E6%96%B9%EF%BC%88n%E4%B8%BA%E5%81%B6%E6%95%B0%EF%BC%89%2C%E6%B1%82%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E7%9A%84%E5%92%8CSn)
数列an的通项公式an=6n-5(n为奇数),an=2的n次方(n为偶数),求数列an的前n项的和Sn
数列an的通项公式an=6n-5(n为奇数),an=2的n次方(n为偶数),求数列an的前n项的和Sn
数列an的通项公式an=6n-5(n为奇数),an=2的n次方(n为偶数),求数列an的前n项的和Sn
1.
当n为偶数时,n=2k
a(2k-1)=6(2k-1)-5)=12k-11
sk=12k(k+1)/2-11k=6k^2-5k
a(2k)=2^(2k)=4^k
tk=4(4^k-1)/3
=(1/3)4^(k+1)-4/3
S(2k)=sk+tk=6k^2-5k+4(4^k-1)/3
=6k^2-5k+(1/3)4^(k+1)-4/3
=(1/3)4^(k+1)+6k^2-5k-4/3
Sn=(1/3)4^(n/2+1)+6(n/2)^2-5n/2-4/3
=(1/3)4^(n/2+1)+(3/2)n^2-(5/2)n-4/3
=(1/3)2^(n+2)+(3/2)n^2-(5/2)n-4/3
2.
当n为奇数时,n=2k-1
a(2k-1)=6(2k-1)-5=12k-11
sk=12k(k+1)/2-11k
=6k^2-5k
a(2k-2)=2^(2k-2)=4^(k-1)
tk=4[4^(k-1)-1]/3
=(1/3)4^k-4/3
S(2k-1)=sk+tk
=6k^2-5k+(1/3)4^k-4/3
=(1/3)4^k+6k^2-5k-4/3
Sn=(1/3)4^[(n+1)/2]+6[(n+1)/2]^2-5(n+1)/2-4/3
=(1/3)2^(n+1)+(3/2)(n+1)^2-(5/2)n-23/6
综上所述
当n为偶数时,Sn=(1/3)2^(n+2)+(3/2)n^2-(5/2)n-4/3
当n为奇数时,Sn=(1/3)2^(n+1)+(3/2)(n+1)^2-(5/2)n-23/6
当n 为奇数时,sn=(1+6n-5)*(n+1)/4 + 4(1-4的(n-1)/2次方)再除以(1-4)
当n为偶数时,sn=(1+6(n-1)-5)*n/4 + 4(1-4的n/2次方)再除以(1-4)