设a1=kc,a2=kc²,a3=kc³(k﹥0,c﹥0),求证:㏒10a1,㏒10a2,㏒10a3成等差数列

设a1=kc,a2=kc²,a3=kc³(k﹥0,c﹥0),求证:㏒10a1,㏒10a2,㏒10a3成等差数列
设a1=kc,a2=kc²,a3=kc³(k﹥0,c﹥0),求证:㏒10a1,㏒10a2,㏒10a3成等差数列

设a1=kc,a2=kc²,a3=kc³(k﹥0,c﹥0),求证:㏒10a1,㏒10a2,㏒10a3成等差数列
㏒10a1＋㏒10a3
＝㏒10(kc)+㏒10(kc³)
=㏒10(k)+㏒10(c)+㏒10(k)+3㏒10(c)
=2㏒10(k)+4㏒10(c)
=2[㏒10(k)+2㏒10(c)]
2㏒10a2
=2㏒10(kc²)
=2[㏒10(k)+2㏒10(c)]
因为,㏒10a1＋㏒10a3＝2㏒10a2
所以,㏒10a1,㏒10a2,㏒10a3成等差数列

log10a1+log10a3=log10a1a3=log10k^2c
^4=log10(a2)^2=2log10a2
所以成等差数列

an=kc^n
log10(an)=log10(kc^n)=log10(k)+n*log10(c)
log10(an)-log10(an-1)=log10(c)
相临项差值是常数，所以log10(a1),log10(a2),log10(a3),...是等差数列

c是常数吧，log10a1+log10a3=log10a1a3=log10（kc²）²=2log10kc²=2log10a2
即证之