Sequence - Geometric Sequences

SOURCE:COMPETITION
Number of Problems: 5. : (Book)

The second and fourth terms of a geometric sequence are $2$ and $6$ . Which of the following is a possible first term?

$extbf{(A) } -sqrt{3} qquad extbf{(B) } -frac{2sqrt{3}}{3} qquad extbf{(C) } -frac{sqrt{3}}{3} qquad extbf{(D) }...$

''>''

The second and fourth terms of a geometric sequence are $2$ and $6$ . Which of the following is a possible first term?

$extbf{(A) } -sqrt{3} qquad extbf{(B) } -frac{2sqrt{3}}{3} qquad extbf{(C) } -frac{sqrt{3}}{3} qquad extbf{(D) }...$

A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

$ext {(A)} 1 qquad ext {(B)} 4 qquad ext {(C)} 36 qquad ext {(D)} 49 qquad ext {(E)} 81$

Positive integers $a$ , $b$ , and $2009$ , with $a<b<2009$ , form a geometric sequence with an integer ratio. What is $a$ ?

$mathrm{(A)} 7qquadmathrm{(B)} 41qquadmathrm{(C)} 49qquadmathrm{(D)} 289qquadmathrm{(E)} 2009$

Consider the set of numbers ${1, 10, 10^2, 10^3, ldots, 10^{10}}$ . The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

$extbf{(A)} 1 qquad extbf{(B)} 9 qquad extbf{(C)} 10 qquad extbf{(D)} 11 qquad extbf{(E)} 101$