Find an explicit formula for the arithmetic sequence -45,-30,-15,0,...
Answers
Answer:
tead of "explicit") go to Sequences as Functions - Recursive.]
Certain sequences (not all) can be defined (expressed) as an "explicit" formula. An explicit formula will create a sequence using n, the number location of each term.
If you can find an explicit formula for a sequence, you will be able to quickly and easily find any term in the sequence simply by replacing n with the number of the term you seek.
An explicit formula designates the nth term of the sequence, as an expression of n (where n = the term's location). It defines the sequence as a formula in terms of n. It may be written in either subscript notation an, or in functional notation, f (n).
Ex1 Sequence: {10, 15, 20, 25, 30, 35, ...}. Find an explicit formula.
This example is an arithmetic sequence(the same number, 5, is added to each term to get to the next term).
Term Number
Term
Subscript Notation
Function Notation
1
10
a1
f (1)
2
15
a2
f (2)
3
20
a3
f (3)
4
25
a4
f (4)
5
30
a5
f (5)
6
35
a6
f (6)
n
dotdotdot
an
f (n)
Explicit Formula:
in subscript notation: an = 5n + 5
in function notation: f (n) = 5n + 5
seqfuncgraph1
This sequence is graphed in the first quadrant. Remember that the domain consists of the natural numbers, {1, 2, 3, ...}, and the range consists of the terms of the sequence. Notice that this sequence has a linear appearance. The rate of change between each of the points is "5 over 1". While the n value increases by a constant value of one, the f (n) value increases by a constant value of 5 (for this graph).
Will arithmetic sequences be linear functions?
scratch head
It is easy to see that the explicit formula works once you are given the formula. Unfortunately, it is not always easy to come up with explicit formulas, when all you have is a list of the terms.
If your sequence is arithmetic, it will help if you look at the pattern of what is happening in the sequence.
funcpatternpict
Explicit formula: f (n) = 10 + 5(n - 1)
If you compare the term number with how many times the common difference, 5, is added, you will see a pattern for an explicit formula:
funcformulaF
To summarize the process of writing an explicit formula for an arithmetic sequence:
1. Determine if the sequence is arithmetic (Do you add, or subtract, the same amount from one term to the next?)
2. Find the common difference. (The number you add or subtract.)
3. Create an explicit formula using the pattern of the first term added to the product of the common difference and one less than the term number.
an= a1 + d (n - 1)
an = the nth term in the sequence
a1 = the first term in the sequence
n = the term number
d = the common difference.
function formula
{10, 15, 20, 25, 30, 35, ...}
first term = 10, common difference = 5
explicit formula: an= 10 + 5(n - 1)
= 10 + 5n - 5 = 5 + 5n or 5n + 5
Now that you have the explicit formula, find the 100th term of this sequence.
Replace n with 100 in the explicit formula: f (n) = 10 + 5(n - 1)
f (100) = 10 + 5(100 - 1) = 10 + 5(99) = 10 + 495 = 505 ANSWER
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Ex2 Sequence: {3, 6, 12, 24, 48, 96, ...}. Find an explicit formula.
This example is a geometric sequence (the same number, 2, is multiplied times each term to get to the next term).
Term Number
Term
Subscript Notation
Function Notation
1
3
a1
f (1)
2
6
a2
f (2)
3
12
a
Step-by-step explanation:
Answer:
all photo answers
Step-by-step explanation:
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