1. Finding a Geometric Progression
MX 233 page 13

2. Find the GP when t4 = 40, t7 = 320 t4 = ar4-1 = ar3 = 40
(ii)
ar6 = 320
ar3 = 40
(ii) ÷ (i)
r3 = 8
r = 2
* odd index, 1 sequence
ar3 = 40
8a = 40
a = 5
Since r = 2 and a = 5, the GP is 5, 10, 20, 40, …

3. Since r = 3 or -3 and a = 2, the two possible GP’s are
Find the GP when t3 = 18, t5 = 162
(i)
t3 = ar3-1 = ar2 = 18
t5 = ar5-1 = ar4 = 162
(ii)
ar4 = 162
ar2 = 18
(ii) ÷ (i)
r2 = 9
r = +3 or -3
* Even index, 2 sequences
ar2 = 18
9a = 18
a = 2
Since r = 3 or -3 and a = 2, the two possible GP’s are
2, 6, 18, 54, 162, … and , -6, 18, -54, 162, …

4. Find GPs for the following conditions
t3 = 1 and t6 = 27
t5 = 5 and t7 = 125
t5 = 48 and t8 = 6
MX233 page 14
The GP is 768, 384, 192, 96, 48, 24, 12, 6, …

5. What happens to the values of tn as n increases?
The Limit of a Sequence
What happens to the values of tn as n increases?
t t t100
4, 3.5, 3.3, 3.25, 3.2, 3.17, …3.05, …….3.02, …….3.01,…
2, 4, 6, 8, 10, 12,…………………40,………..90,……….200,…
-3, 3, -3, 3, -3, 3, ………………3,…………-3, ………..3, ….
-1.2, -1.44, , ,… ,… ,… ,.
1. As n increases the value of tn gets nearer and nearer to 3
2. As n increases the value of tn increases.
3. Throughout the sequence, the value of tn is either 3 or -3.
We say the sequence oscillates between 3 and -3
4. As n increases the value of tn becomes more negative.

6. For each of the following sequences T1 to T6, T45 and T100 are given
For each of the following sequences T1 to T6, T45 and T100 are given. What happens to the value of Tn as n increases in each case?
0.1, 0.4, 0.9, 1.6, 2.5, 3.6, …40,…202.5,…1000,…
0.91, 0.83, 0.75, 0.68, 0.62, 0.56,…0.15,…0.014,… ,…
-1, 1, -1, 1, -1, 1,…1,…-1,…1,…
1, 1.33, 1.5, 1.6, 1.67, 1.71,…1.90,…1.96,…1.98,…
-5, -13, -21, -29, -37, -45,…-157,…-357,…-797,… .
3.6, 2.5, 3.3, 2.8, 3.1, 2.95,…2.98,…3.005,…2.9973,…
0.83, -0.69, 0.58, -0.48, 0.40, -0.33,…-0.026,… ,… ,…

7. “the limit of tn, as n tends to infinity, is 3”
For the sequence
4, 3.5, 3.3, 3.25, 3.2, 3.17, …3.05, …….3.02, …….3.01,…
we found as n increased the value of tn got nearer and nearer to 3
“the limit of tn, as n tends to infinity, is 3”

8. For any sequence with general term tn
If, as n increases, tn approaches a value ℓ more and more closely
and the sequence has a limit ℓ
If, as n increases, | tn | increases without limit, or tn oscillates,
and the sequence has no limit.