Learn how to find explicit formulas for arithmetic sequences. For example, find an explicit formula for 3, 5, 7,...
Log in 😊 7 years agoPosted 7 years ago. Direct link to 😊's post “what dose it mean to crea...” what dose it mean to create an explicit formula for a geometric • (24 votes) Kim Seidel 7 years agoPosted 7 years ago. Direct link to Kim Seidel's post “An explicit formula direc...” An explicit formula directly calculates the term in the sequence that you want. (16 votes) Shelby Anderson 7 years agoPosted 7 years ago. Direct link to Shelby Anderson's post “Can you add a section on ...” Can you add a section on Simplifying Geometric and arithmetic equations? I have a test on this tomorrow, and there isn't a section to help me study... Wish me luck I guess :~: • (12 votes) Franscine Garcia 7 years agoPosted 7 years ago. Direct link to Franscine Garcia's post “What's the difference bet...” What's the difference between this formula and a(n) = a(1) + (n - 1)d? Is one better or something? Tzarinapup 7 years agoPosted 7 years ago. Direct link to Tzarinapup's post “The reason we use a(n)= a...” The reason we use a(n)= a+b( n-1 ), is because it is more logical in algebra (9 votes) Siegrid Pregartner 6 years agoPosted 6 years ago. Direct link to Siegrid Pregartner's post “To find the common differ...” 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? • (4 votes) Tim Nikitin 6 years agoPosted 6 years ago. Direct link to Tim Nikitin's post “Your shortcut is derived ...” 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: 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". (8 votes) sop 10 months agoPosted 10 months ago. Direct link to sop's post “The first term of an arit...” The first term of an arithmetic sequence is • (2 votes) Shayne Lee 10 months agoPosted 10 months ago. Direct link to Shayne Lee's post “the common difference is ...” 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 (6 votes) 19.amber.broyhill 7 years agoPosted 7 years ago. Direct link to 19.amber.broyhill's post “what is the recursive for...” what is the recursive formula for airthmetic formula • (3 votes) ca9034266 a year agoPosted a year ago. Direct link to ca9034266's post “do we need to know arithm...” do we need to know arithmetic sequences for the SAT? • (2 votes) Lincoln Earley 10 months agoPosted 10 months ago. Direct link to Lincoln Earley's post “It'll probably help. Most...” 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. (3 votes) Dzeerealxtin 7 years agoPosted 7 years ago. Direct link to Dzeerealxtin's post “Determine the next 2 term...” Determine the next 2 terms of this sequence • (0 votes) Timber Lin 7 years agoPosted 7 years ago. Direct link to Timber Lin's post “warning: long answerthis...” warning: long answer simplify: 2=a+b+c solve this using any method, but i'll use elimination: 5=4a+2b+c then subtract the 2 equations just produced: so the equation becomes y=1x^2+0x+1, or y=x^2+1 (8 votes) Alex T. 8 years agoPosted 8 years ago. Direct link to Alex T.'s post “It seems to me that 'expl...” 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? • (2 votes) Ken Burwood 7 years agoPosted 7 years ago. Direct link to Ken Burwood's post “m + Bn and A + B(n-1) are...” 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: Turn it into an explicit formula by taking the initial term's value and adding it to B times the integer (n-1): (2 votes) louisaandgreta 3 years agoPosted 3 years ago. Direct link to louisaandgreta's post “How do you algebraically ...” How do you algebraically get • (1 vote) Jerry Nilsson 3 years agoPosted 3 years ago. Direct link to Jerry Nilsson's post “3 + 2(𝑛 − 1)= [distribu...” 3 + 2(𝑛 − 1) (3 votes)Want to join the conversation?
A recursive formula calculates each term based upon the value of the prior term. So, it usually takes more steps.
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").
5 and the tenth term is 59
How do we make sure the common difference is positive 6 and not negative 6?
2,5,10,17.. then write the explicit form
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
5=4a+2b+c
10=9a+3b+c
10=9a+3b+c
-(5=4a+2b+c)
5=5a+b (equation 3 - equation 2)
-(2=a+b+c)
3=3a+b (equation 2 - equation 1)
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
btw you can check (4,17) to make sure it's righta(1) = Aa(n) = a(n-1) + B (here a(n-1) is this function for the previous term, not multiplication)a(n) = A + B(n-1) (here B(n-1) is multiplication, not a function)
5+2(n-2) from
the standard form 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)
