There are two positive numbers that may be inserted between 3 and 9 such that the first three numbers are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is:
Note: There are two other numbers that work, but they’re not both positive. If you go about this problem in a suitably erudite fashion, you’ll turn up this alternative solution too.
Continue these sequences. Explain how each one works.
| a. | 0 | 1 | 3 | 6 | 7 | 9 | 12 | 13 | ___ | ___ | ___ |
| b. | 1 | 2 | 3 | 4 | 9 | 8 | 27 | 16 | ___ | ___ | ___ |
| c. | 0 | 4 | 8 | 4 | 8 | 12 | 6 | 10 | ___ | ___ | ___ |
| d. | 1 | 2 | 3 | 3 | 6 | 9 | 9 | ___ | ___ | ___ | ___ |
| e. | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | ___ | ___ | ___ |
| f. | 0 | 1 | 1 | 2 | 4 | 7 | 13 | 24 | ___ | ___ | ___ |
Here is a rather unusual sequence: 125, 59, 42, 34, 14, 4, ____. Can you finish the sequence?
Find an integer n such that (1 + 2 + 3 + . . . + n)/3n = 36.
The x-axis and the three lines x = 1, x = 4 and y = mx + 4 form a trapezoid. If the area of the trapezoid is 7, then what is m?
Given: lines y = 3x + 1, y = -2x + 5, and a third line L1: y = mx + c with m > 0 and c a constant. L1 moves parallel to itself intersecting the two given lines in points P1 and P2 respectively. The locus of mid-points of segment P1P2 is line L2. Find the slope of L1 so that L2 has a slope that is undefined.
The arch of a stone bridge has the form of a semi-ellipse. The straight horizontal span is 40 meters and the maximum height between the arch and the span is 10 meters. Find the height, in meters, of the arch 10 meters from one end of the span.
If w is one of the imaginary roots of the equation x3 = 1,
then the quartic (1 – w + w2)(1 + w – w 2) is equal to:
One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form
