gre数学”算数“基础要牢靠2020-01-11 19:03 来源：未知
1.Sum of Arithmetic Progression(等差数列求和)
The sum of n-numbers of an arithmetic progression is given by
where x is the first number and d is the constant increment.
sum of first 10 positive odd numbers:10*1+2*10*9/2=10+90=100
sum of first 10 multiples of 7 starting at 7： 10*7+7*10*9/2=70+315=385
For a descending AP the constant difference is negative.
Average of n numbers of arithmetic progression (AP) is the average of the smallest and the largest number of them. The average of m number can also be written as x + d(m-1)/2.
The average of all integers from 1 to 5 is (1+5)/2=3
The average of all odd numbers from 3 to 3135 is (3+3135)/2=1569
The average of all multiples of 7 from 14 to 126 is (14+126)/2=70
Make sure no number is missing in the middle.
With more numbers, average of an ascending AP increases.
With more numbers, average of a descending AP decreases.
AP:numbers from sum
given the sum s of m numbers of an AP with constant increment d, the numbers in the set can be calculated as follows:
the first number x = s/m - d(m-1)/2,and the n-th number is s/m + d(2n-m-1)/2.
if the sum of 7 consecutive even numbers is 70, then the first number x = 70/7 - 2(7-1)/2 = 10 - 6 = 4.
the last number (n=m=7)is 70/7+2(2*7-7-1)/2=10+6=16.the set is the even numbers from 4 to 16.
given the first number x, it is easy to calculate other numbers using the formula for n-th number: x+(n-1)
AP:numbers from average
all m numbers of an AP can be calculated from the average. the first number x = c-d(m-1)/2， and the n-th number is c+d(2n-m-1)/2, where c is the average of m numbers.
if the average of 15 consecutive integers is 20, then the first number x=20-1*(15-1)/2=20-7=13 and the last number (n=m=15) is 20+1*(2*15-15-1)/2=20+7=27.
if the average of 33 consecutive odd numbers is 67, then the first number x=67-2*(33-1)/2=67-32=35 and the last number (n=m=33) is 67+2*(2*33-33-1)/2=67+32=99.
sum of the m numbers is c*m,where c is the average.
3.Sequence of Numbers(序列)
A sequence is a set of numbers that follow a fixed pattern.The fixed pattern can be expressed by an equation or by a property.
A set of consecutive integers: 1,2,3,4,5(Fixed gap)
A set of consecutive even numbers:4,6,8,10,12 (Fixed gap)
A set of consecutive prime: 2,3,5,7,11(Fixed gap)
A set of consecutive power of 2:4,8,16,32,64(Fixed gap)
A sequence can be in ascending or descending order.