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20 17 arithmetic progression's formula must be tested in mathematics for the college entrance examination.
Arithmetic progression is a common series. If a series starts from the second term and the difference between each term and its previous term is equal to the same constant, this series is called arithmetic progression. The following is the information about arithmetic progression's formula that I compiled for you, which is required for the 20 17 college entrance examination. I hope it helps you.

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).