Question 1

what dose it mean to create an explicit formula for a geometric

Accepted Answer

An explicit formula directly calculates the term in the sequence that you want.
A recursive formula calculates each term based upon the value of the prior term.  So, it usually takes more steps.

Question 2

What's the difference between this formula and a(n) = a(1) + (n - 1)d? Is one better or something?

Accepted Answer

The reason we use a(n)= a+b( n-1 ), is because it is more logical in algebra

Question 3

To find the common difference between two terms, is taking the difference and dividing by the number of terms a viable workaround?

For example, the first term is 5 and the tenth term is 59. So take the difference, 59-5=54, then divide by the number of terms between them, which is 5. And the average difference would be 6. Does this always work?

Accepted Answer

Your shortcut is derived from the explicit formula for the arithmetic sequence like 5 + 2(n – 1) = a(n). Plug your numbers into the formula where x is the slope and you'll get the same result: 
5 + x(10 – 1) = 59
5 + 9x = 59
9x = 54
x = 6
To find the slope, you take the difference between the 10th and the 1st term and divide it by the "# of additions" or by the difference between the 10th term and the 1st term.
So your shortcut works always, because it's the same thing as the explicit formula (5 + 6(10 – 1) = 59). Basically, it says to get the 10th number in the sequence, you have to start from the base of 5 then add 6 (the slope) to it not once, but in this case 9 times ("# of additions").

P.S. You have a typo in "divide by the number of terms between them, which is 5". It should be 9, not 5. Also the "average difference" should be called "common difference" or "slope".

Question 4

The first term of an arithmetic sequence is 
5 and the tenth term is 59
How do we make sure the common difference is positive 6 and not negative 6?

Accepted Answer

the common difference is constant so we know the sequence will either get bigger or smaller, the tenth term is bigger than the first term so we know it will be positive

Question 5

do we need to know arithmetic sequences for the SAT?

Accepted Answer

It'll probably help. Most of this kind of stuff is less for real world applications but for creating foundations of understanding for later skills that might require a foundation in understanding and representing sequences correctly. You never know what's ahead.

Question 6

It seems to me that 'explicit formula' is just another term for iterative formulas, because both use the same form. Is this true? And is there another term for formulas using the _m_ + _Bn_ form as opposed to the _A_ + _B(n-1)_ form or are they both referred to as explicit formulas?

Accepted Answer

m + Bn and A + B(n-1) are both equivalent explicit formulas for arithmetic sequences.  A + B(n-1) is the standard form because it gives us two useful pieces of information without needing to manipulate the formula (the starting term A, and the common difference B).

An explicit formula isn't another name for an iterative formula.  Even though they both find the same thing, they each work differently--they're NOT the same form.

In the iterative formula, "a(n-1)" means "the value of the (n-1)th term in the sequence", this is not "a times (n-1)."

In the explicit formula "d(n-1)" means "the common difference times (n-1), where n is the integer ID of term's location in the sequence."

Thankfully, you can convert an iterative formula to an explicit formula for arithmetic sequences. Converting is usually less work.

*Take the iterative formula*:
`a(1) = A`
`a(n) = a(n-1) + B` (here a(n-1) is this function for the previous term, not multiplication)

*Turn it into an explicit formula* by taking the initial term's value and adding it to B times the integer (n-1):
`a(n) = A + B(n-1)` (here B(n-1) is multiplication, not a function)

Question 7

How do you algebraically get 
5+2(n-2) from 
the standard form 3+2(n-1)?

Accepted Answer

3 + 2(𝑛 − 1)
= [distribute the 2] = 3 + 2𝑛 − 2
= [add 2 and subtract 2] = 3 + 2 + 2𝑛 − 2 − 2 = 5 + 2𝑛 − 4
= [factor out 2] = 5 + 2(𝑛 − 2)

Question 8

Determine the next 2 terms of this sequence
2,5,10,17.. then write the explicit form

Accepted Answer

warning: long answer
this isn't an arithmetic ("linear") sequence because the differences between the numbers are different (5-2=3, 10-5=5, 17-10=7)
however, you might notice that the differences of the differences between the numbers are equal (5-3=2, 7-5=2). that means the sequence is quadratic/power of 2.
since the sequence is quadratic, you only need 3 terms.
let x=the position of the term in the sequence
let y=the value of the term
the 1st term is 2, so x=1 and y=2
the 2nd term is 5, so x=2 and y=5
the 3rd term is 10, so x=3 and y=10
the function is y=ax^2+bx+c, so plug in each point to solve for a, b, and c.
(1,2): 2=a(1^2)+b(1)+c
(2,5): 5=a(2^2)+b(2)+c
(3,10): 10=a(3^2)+b(3)+c

simplify: 2=a+b+c
5=4a+2b+c
10=9a+3b+c

solve this using any method, but i'll use elimination:
10=9a+3b+c
-(5=4a+2b+c)
5=5a+b  (equation 3 - equation 2)

5=4a+2b+c
-(2=a+b+c)
3=3a+b  (equation 2 - equation 1)

then subtract the 2 equations just produced:
5=5a+b
-(3=3a+b)
2=2a
that means a=1.
substitute a=1 into 3=3a+b: 3=3+b, b=0.
substitute a and b into 2=a+b+c: 2=1+0+c, c=1

so the equation becomes y=1x^2+0x+1, or y=x^2+1
btw you can check (4,17) to make sure it's right

$n$ ‍	Calculation for the $n^{th}$ ‍ term
$1$ ‍	$5$ ‍	$= 5 + 0 \cdot 3 = 5$ ‍
$2$ ‍	$5 + 3$ ‍	$= 5 + 1 \cdot 3 = 8$ ‍
$3$ ‍	$5 + 3 + 3$ ‍	$= 5 + 2 \cdot 3 = 11$ ‍
$4$ ‍	$5 + 3 + 3 + 3$ ‍	$= 5 + 3 \cdot 3 = 14$ ‍
$5$ ‍	$5 + 3 + 3 + 3 + 3$ ‍	$= 5 + 4 \cdot 3 = 17$ ‍

Algebra 1

Course: Algebra 1 > Unit 9

Explicit formulas for arithmetic sequences

How explicit formulas work

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Writing explicit formulas

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Equivalent explicit formulas

A common misconception

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Challenge problems

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