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.