What would be the 50th term of the series having the generalised formula 5m +2?
Answers
Answer:
This is the formula that will be used when we find the general (or nth) term of an arithmetic sequence.
EXAMPLE 1: Find the general (or nth) term of the arithmetic sequence :
2, 5, 8, ........
Solution: Since we are told that the sequence is arithmetic we know that the difference between ANY two consecutive terms is a constant, d. Therefore, d = 5 - 2 = 3.
Plugging d = 3 along with a1 = 2 into the general formula for an arithmetic sequence we get the following:
an = 2 + (n - 1) ( 3)
an = 2 + 3n - 3
General term: an = 3n - 1
We can also determine the general term of an arithmetic sequence when we know that the sequence is arithmetic and we know two of the terms.
EXAMPLE 2: Find the general (or nth) term of the arithmetic sequence that has a 5th term of 15 and 10th term of 12.
Solution: We again begin with the formula for the general term of an arithmetic sequence
Equation 1: a5 = a1 + 4d = 15
Equation 2: a10 = a1 + 9 d = 12
This is a system of two equations and two unknowns which we will use to find a1 & d.
Restatement of Eq1: a1 = 15 - 4d
Substitute Eq1 into Eq2: (15 - 4d) + 9d = 12
15 + 5d = 12
5d = - 3
d = -3/5
Substituting d = - 3/5 into Eq1 yields a1 = 15 - 4(-3/5) = 87/5
General term: an = 87/5 + (n - 1) (-3/5)
an = 18 + (-3/5)n
The two examples just covered refer to finding the general term when we are guaranteed up front that the sequence is arithmetic. Knowing that the sequence is arithmetic allows us to use the pattern of an arithmetic sequence in order to find the general term.
If we did not know something of the pattern then our task of finding the general term would be much more difficult. On the flip side, when we need to PROVE that a sequence is arithmetic we must show that the sequence follows the pattern of an arithmtic sequence. For a review of how to prove that a squence is arithmetic, read through the Supplementary Reading on that topic.
We now turn our attention to geometric sequences.
Finding the nth term of a geometric sequence
A geometric sequence is one in which the ratio of consecutive terms is a constant. We often symbolize this constant ratio by r. To generate the terms of a geometric sequence we just keep multiplying the last known term by the same number, r. That is, if a1 represents the first term and r is the common ratio then
a2 = a1 r
a3 = a2 r = (a1 r) r = a1 r2
a4 = a3 r = (a1 r2) r = a1 r3
Capturing this pattern in alegbra,
we write the general (or nth) term of a geometric sequence as:
an = a1 rn - 1
This is the formula that will be used when we find the general (or nth) term of a geometric sequence.
EXAMPLE 3: Find the general term of the geometric sequence: 8, 4, 2, ......
Solution: Since we are told that the sequence is geometric we know that the ratio of ANY two consecutive terms is a constant, r. Therefore, r = 4/8 = 1/2.
Plugging r = 1/2 along with a1 = 8 into the general formula for a geometric sequence we get the following:
an = 8 (1/2) n - 1
General term: an = 8 (1/2) n - 1
As you might suspect, it is also possible to find the general term of a geometric sequqnce once we know two terms of the sequence. The steps mirror Example 2 so we won't repeat all of that again (unless you email me and tell me that you MUST have all of the details).
Find the summation form of a series
Recall that a series is simply the sum of the terms of a sequence. If the sequence to be added is either arithmetic or geometric then we can use Examples 1 & 3 to write the series in summation form.
EXAMPLE 4: Write the following series in summation form:
2 + 5 + 8 + 11 + 14 + 17 + 20 + 23
Solution: Notice that the sequence of terms is arithmetic. In fact, it is the sequence discussed in Example 1 where we found that the general term is an = 3n -1. We want to add the first 8 terms of this sequence.
Summation form:
Here is another example of writing a series in summation notation. In this problem the series is neither arithmetic nor geometric but we still can use arithmetic and geometric patterns to express the general term.
EXAMPLE 5: Write the following series in summation form:
2/2 + 5/4 + 8/8 + 11/16 + 14/32 + 17/64 + 20/128
Solution: This sequence is neither arithmetic or geometric but the numerator and denominator are!
Numerator: 2, 5, 8, 11, 14, 17, 20
Denominator: 2, 4, 8, 16, 32, 64, 128
The numerator is the same arithmetic sequence that we have encountered in Examples 1 & 4 that has a general term of an = 3n - 1
The denominator is a geometric sequence with a1 = 2 and r = 2. Plugging those values into the general form of the geometric sequence (as done in Example 2) we find that the general term for the denominator is an = 2 (2)n-1 = 2 n
Dividing the numerator by the denominator we get the following general term:
(3n-1) / (2n)
Since we want to add the first seven terms of the sequence we get the following:
Summation form:
At this point you might want to go back to your text and try some of the problems related to this topic. As always, if you have questions about what is written here, if you find mistakes, or if you have comments, please send them to 54
Answer:
252
Step-by-step explanation:
In the A.P ,
General Formula = an = a + (n – 1) × d
Here, it is given 5m + 2
Here m will the number of terms.
Applying 50 to m, we get
a50 = 2 + 5 (50) = 252