Find the ratio of consecutive pair of terms in the following sequences.
1,2,3,4,8,16
Answers
Answer:
Finding Common Ratios
The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.
A GENERAL NOTE: DEFINITION OF A GEOMETRIC SEQUENCE
A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If \displaystyle {a}_{1}a
1
is the initial term of a geometric sequence and \displaystyle rr is the common ratio, the sequence will be
{
a
1
,
a
1
r
,
a
1
r
2
,
a
1
r
3
,
…
}
.
HOW TO: GIVEN A SET OF NUMBERS, DETERMINE IF THEY REPRESENT A GEOMETRIC SEQUENCE.
Divide each term by the previous term.
Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.
Answer:
geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... is geometric, because each step multiplies by two; and 81, 27, 9, 3, 1, \frac{1}{3}
3
1
,... is geometric, because each step divides by 3.
The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (that is, if you find the ratio of) successive terms, you'll always get this common value.