數(shù)列{an}的前n項(xiàng)和Sn,已知對(duì)任意的n∈N*,點(diǎn)(n,Sn)均在函數(shù)y=ax2+x(a∈N*)的圖象上,則( )
A.a(chǎn)與an的奇偶性相同
B.n與an的奇偶性相同
C.a(chǎn)與an的奇偶性相異
D.n與an的奇偶性相異
【答案】分析:本題主要考查數(shù)列通項(xiàng)an與前n項(xiàng)和Sn之間的關(guān)系及函數(shù)解析式.首先將點(diǎn)代入函數(shù)解析式確定an與Sn的關(guān)系,然后利用通過(guò)此式求得通項(xiàng)公式,最后分析n與an的奇偶性.本題易忽視判斷a與a1奇偶性,即忽視a1與S1的關(guān)系.
解答:解:∵對(duì)任意的n∈N*,點(diǎn)(n,Sn),均在函數(shù)y=ax2+x(a∈N*)的圖象上
∴Sn=an2+n,
當(dāng)n=1時(shí),a1=S1=a+1,
當(dāng)n≥2時(shí),an=Sn-Sn-1=an2+n-[a(n-1)2+(n-1)]=2an-a+1,
當(dāng)n=1時(shí),2an-a+1=a1.
∴an=2an-a+1=(2n-1)a+1,
∴a與an的奇偶性相異.
故選C.
點(diǎn)評(píng):本題考查數(shù)列與函數(shù)的綜合,解題時(shí)要認(rèn)真審題,注意挖掘題設(shè)中的隱含條件,合理地進(jìn)行等價(jià)轉(zhuǎn)化.