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Summary of 3 pages for the course Mathematics at 12th Grade (will help you)
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The Geometric Sequence and Series1. 16 + + 8 + 4 + . . . are the first three terms in a geometric series.
a) Write down the nth term in this series.
b) How many terms should be added to get a sum that exceeds 31?
c) Calculate the sum to infinity, if it exists.
d) Write the series in Sigma notation.
2. The sequence 2
1 ; − 15 ; − 254 ; . . . continues indefinitely in the same manner.
(a) Determine the nth term of this sequence.
(b) Which term will be the first to have a numerator smaller than -56?
and what will the value of the denominator be for this term?
3. (
In a geometric sequence, Sn = p 1 − ( 12 )
n
) and S = 10. Find the value of p.
4. In a geometric sequence, Tn = ( 4 x − 1) .
n
(a) Find the value(s) of x for which S exists.
(b) Choose any value for x on this interval and calculate S .
5. Show that the sequence 1 ; x + 1 ; x − 3 ; . . . is not a geometric sequence.
6. The sum of the second and the third terms of a G.P. is 280, and the sum of the 5th and the 6th terms
is 4375. Find the 4th term.
7. The first three terms of a geometric sequence is 4 ; -2; 1.
1
Which term in this sequence will have a value of 64
?
8.
Twenty water tanks are decreasing in size in such a way that the volume of each tank is half the volume of
the previous tank. The first tank is empty, but the other tanks are full to capacity. Will the first tank be able
to hold all the water of the other 19 tanks?
Number Patterns: Mixed
9. The following sequence is given: 3 ; 3 ; 9 ; 6 ; 15 ; 12 ; . . .
(a) Calculate T52 − T51.
(b) Will this sequence always be divisible by 3? Motivate.
(c) Calculate the sum of the first 50 terms of this sequence.
10. The arithmetic mean between two numbers is 39 and the geometric mean is15.
Find the numbers.
11. Write the following series in sigma notation: (1 2) + ( 5 6) + ( 9 10 ) + (13 14 ) + . . . + (81 82 ) .
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