Geometric sequences

1003158501

Level: 
C
Let \( 3 \) numbers be \( 3 \) consecutive terms of a geometric sequence with the common ratio \( q=4 \). If we increase the second number by \( 9 \), we get \( 3 \) consecutive arithmetic sequence terms. Find the first number.
\( 2 \)
\( 4 \)
\( 8 \)
\( 16 \)
\( 32 \)

1003158502

Level: 
C
Let there be two unknown positive numbers between numbers \( 12 \) and \( 54 \). The first three numbers of these four form \( 3 \) consecutive arithmetic sequence terms, and the last \( 3 \) numbers form three consecutive terms of a geometric sequence. Find the smaller of the two unknown numbers.
\( 24 \)
\( 36 \)
\( 15 \)
\( 20 \)
\( 32 \)

1003158503

Level: 
C
Let there be four numbers, such that the first three numbers form consecutive arithmetic sequence terms with a difference of \( d=-6 \), and the last three numbers form consecutive terms of a geometric sequence with the common ratio \( q=\frac23 \). Find the fourth number.
\( 8 \)
\( 18 \)
\( 12 \)
\( -24 \)
\( -4 \)

1003158504

Level: 
C
Let there be three numbers that are three consecutive arithmetic sequence terms with a difference of \( d=3 \). If the third number is decreased by \( \frac32 \), we get \( 3 \) consecutive terms of a geometric sequence. Find the third number (of an arithmetic sequence).
\( 0 \)
\( 3 \)
\( -3 \)
\( \frac32 \)
\( -\frac32 \)

1003158505

Level: 
C
Let there be three numbers that are \( 3 \) consecutive arithmetic sequence terms. Their sum is \( 9 \). If the first number is divided by \( -3 \), we get \( 3 \) consecutive terms of a geometric sequence. Find the largest number of the given three.
\( 9 \)
\( 3 \)
\( 12 \)
\( 6 \)
\( 4 \)

1003158506

Level: 
C
Let there be the first nine terms of an arithmetic sequence with the first term \( a_1=1 \) and the difference \( d=1 \). Suppose we are creating ordered triples of \( 3 \) different numbers out of the given \( 9 \) numbers so that they form \( 3 \) consecutive terms of a geometric sequence. How many of such triples can be created?
\( 8 \)
\( 6 \)
\( 4 \)
\( 3 \)
\( 9 \)

1003158507

Level: 
C
Let there be a row of five yellow cubes lying side by side. The first cube has an edge length of \( 100\,\mathrm{cm} \) and each next cube has an edge length by \( 10\,\mathrm{cm} \) shorter than the previous one. Let there be a second row of five blue cubes lying side by side. The first blue cube has an edge length of \( 100\,\mathrm{cm} \) and each next cube has an edge length by \( 10\% \) smaller than the previous one. What is the difference between the length of these two rows?
\( 9.51\,\mathrm{cm} \)
\( 34.51\,\mathrm{cm} \)
\( 0\,\mathrm{cm} \)
\( 20\,\mathrm{cm} \)
\( 20.51\,\mathrm{cm} \)

1003170601

Level: 
C
How many numbers do we need to insert between the numbers \( 6 \) and \( 1\,458 \) so that the inserted numbers with the given two numbers are consecutive terms of a geometric sequence? The sum of all numbers inserted must be \( 720 \).
\( 4 \)
\( 6 \)
\( 3 \)
\( 5 \)
\( 7 \)

1003170602

Level: 
C
The lengths of a cuboid edges form \( 3 \) consecutive terms of a geometric sequence. The volume of the cuboid is \( 140\,608\,\mathrm{cm}^3 \), the sum of its shortest and its longest edge is \( 221\,\mathrm{cm} \). Find the length of its shortest edge.
\( 13\,\mathrm{cm} \)
\( 52\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 208\,\mathrm{cm} \)
\( 0.25\,\mathrm{cm} \)

1003170603

Level: 
C
Between the roots of the equation \( 9x^2+130x-75=0 \) insert two numbers so that the roots and the new numbers form \( 4 \) consecutive terms of a geometric sequence. What is the smaller of the two inserted numbers?
\( -\frac53 \)
\( \frac59 \)
\( -\frac59 \)
\( \frac53 \)
\( -5 \)