已知等差数列an和正项等比数列bna1=1已知等差数列an和正项等比数列bna1=1,a1=b1=1a3+a5+a7=9,a7是b3和b7等比中项,求cn=2an*bn^2,求数列cn前n项和Tn

Question

已知等差数列an和正项等比数列bna1=1
已知等差数列an和正项等比数列bna1=1,a1=b1=1
a3+a5+a7=9,a7是b3和b7等比中项,求cn=2an*bn^2,求数列cn前n项和Tn

Answer 1

∵等差数列an a1=1 a3+a5+a7=9 ∴a3＋a7＝2a5 ∴3a5＝9 ∴a5＝3
∴a5＝a1＋4d＝1＋4d＝3 ∴d＝1/2 ∴an＝(n＋1)/2
∴a7＝4
∵ 正项等比数列bn b1=1 a7是b3和b7等比中项
∴b3²＝a7×b7 ∴(b1q²)²＝4b1q^6 ∴q²＝4 ∴q＝2 ∴bn＝2^(n－1)
∴cn＝2an×bn²＝4×[2^(n－1)]²＝4^n
∴Tn＝4×(1－4^n)/(1－4)＝4×(4^n－1)/3