Knowledge points of senior high school mathematics: arithmetic progression formula
Arithmetic progression formula an=a 1+(n- 1)d
A 1 is the first term, an is the general term formula of the nth term, and d is the tolerance.
The first n terms and formulas are: Sn=na 1+n(n- 1)d/2.
Sn=(a 1+an)n/2
If m+n=p+q, then: am+an=ap+aq exists.
If m+n=2p, then: am+an=2ap.
All the above n.m.p.q are positive integers.
Analysis: the value of the nth item an= the first item+(number of items-1)? tolerate
The sum of the first n terms Sn= the first term? N+ project number (project number-1) tolerance /2
Tolerance d=(an-a 1)? (n- 1)
Number of items = (last item-first item)? Tolerance+1
When the series is odd, the sum of the first n terms = the middle term? number of terms
If the series is even, find the sum of the first term and the last term and divide it by 2.
The arithmetic mean formula 2an+ 1=an+an+2 where {an} is arithmetic progression.
General formula: tolerance? Item Quantity+First Item-Tolerance
High school mathematics knowledge points: summation formula of arithmetic sequence
If the first term of arithmetic progression is a 1 and the last term is, then the expression of arithmetic progression sum is:
S=(a 1+an)n？ 2
That is (the first item+the last item)? Number of projects? 2
The first n terms and formulas
Note: n is a positive integer (equivalent to the sum of items in n arithmetic)
Arithmetic progression's summation of the first n terms is actually a wonderful application of the trapezoidal formula:
The upper bottom is the first term of a 1, the lower bottom is a 1+(n- 1)d, and the height is n.
That is [a1+a1+(n-1) d] * n/2 = {a1n+n (n-1) d}/2.
Knowledge points of high school mathematics: reasoning process
Let the first item be, the last item be, the number of items be, the tolerance be, and the sum of the previous items be, then there are:
What time? 0, Sn is a quadratic function of n, and (n, Sn) is a set of isolated points on the image of quadratic function. Using its geometric meaning, we can find the maximum value of the first n terms and Sn.
Note: Formulas 1, 2 and 3 are actually equivalent, and the tolerance in Formula 1 is not necessarily required to be equal to one.
Sum derivative
Proof: from the meaning of the question:
Sn=a 1+a2+a3+.。 . +an①
Sn=an+a(n- 1)+a(n-2)+.。 . +a 1②
①+② Obtain:
2sn = [a1+an]+[a2+a (n-1)]+[a3+a (n-2)]+...+[a1+an] (when n is even).
sn = {[a 1+an]+[a2+a(n- 1)]+[a3+a(n-2)]+...+[a 1+an]}/2
Sn=n(A 1+An)/2 (a 1, An, which can be expressed in the form of A 1+(N- 1) D, we can find that the numbers in brackets are all fixed values, that is, (a1+).
basic recipe
Equation Sn=(a 1+an)n/2.
Arithmetic progression's summation formula
sn = na 1+n(n- 1)d/2； (d is the tolerance)
sn = An2+Bn； A=d/2，B=a 1-(d/2)
Sum is serial number.
A 1 item 1
Final answer
Tolerance d
The term quantity
Representation method
Arithmetic progression's basic formula:
The last item = the first item+(item number-1)? tolerate
Number of items = (last item-first item)? Tolerance+1
The first item = the last item-(item number-1)? tolerate
Sum = (first item+last item)? Number of projects? 2
Poor: the first item+the number of items? (project number-1)? Tolerance? 2
explain
The last item: the last digit
Item 1: the first digit
Number of items: How many digits does a * * * have?
Sum: Find the sum of a * * * number
General term formula in this paragraph
The first item =2? And then what? Number of Items-Last Item
The last item =2? And then what? Number of items-first item
The last item = the first item+(item number-1)? Tolerance: a 1+(n- 1)d
Project number = (last project-first project)/tolerance+1:n = (an-a1)/d+1.
Tolerance = d=(an-a 1)/n- 1
For example: 1+3+5+7+99 The tolerance is 3- 1.
Extending a 1 to am is:
d=(an-am)/n-m
Basic attribute
What if m, n, p and q? ordinary
① if m+n=p+q, then am+an=ap+aq.
② if m+n=2q, then am+an=2aq (arithmetic mean).