已知等比数列bn与数列an满足bn=2^an1)判断an是什么数列,并证明2)若a8+a13=1/2,求b1b2····b20

Question

已知等比数列bn与数列an满足bn=2^an
1)判断an是什么数列,并证明 2)若a8+a13=1/2,求b1b2····b20

Answer 1

b(n)=bq^(n-1),n=1,2,...bq不等于0.
b(n) = 2^a(n),
ln[b(n)] = a(n)ln(2),a(1) = ln[b(1)]/ln(2) = ln(b)/ln(2).
q = b(n+1)/b(n) = 2^a(n+1)/2^a(n) = 2^[a(n+1)-a(n)],
ln(q) = [a(n+1)-a(n)]ln(2),
a(n+1)-a(n)=ln(q)/ln(2).
{a(n)}是首项为a(1)=ln(b)/ln(2),公差为ln(q)/ln(2)的等差数列.
a(n) = ln(b)/ln(2) + ln(q)/ln(2)[n-1],n = 1,2,...
1/2 = a(8) + a(13) = ln(b)/ln(2) + 7ln(q)/ln(2) + ln(b)/ln(2) + 12ln(q)/ln(2) = 2ln(b)/ln(2) + 19ln(q)/ln(2) = [2ln(b) + 19ln(q)]/ln(2)
= ln[b^2*q^(19)]/ln(2).
ln[2^(1/2)] = ln(2)/2 = ln[b^2*q^(19)],
b^2*q^(19) = 2^(1/2).
b(1)b(2)...b(20) = b*bq*...*bq^(19) = b^(20)*q^[190] = [b^2*q^19]^(10)
= [2^(1/2)]^(10) = 2^5 = 32.