Each Xian or Kai will have 128 stamps after 6 months.
Firstly , let us learn about types of sequence in mathematics.
Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.
[tex]\boxed {T_n = a + (n-1)d}[/tex]
[tex]\boxed {S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]
Tn = n-th term of the sequence
Sn = sum of the first n numbers of the sequence
a = the initial term of the sequence
d = common difference between adjacent numbers
Geometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ratio.
[tex]\boxed {T_n = a ~ r^{n-1}}[/tex]
[tex]\boxed {S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]
Tn = n-th term of the sequence
Sn = sum of the first n numbers of the sequence
a = the initial term of the sequence
r = common ratio between adjacent numbers
Let us tackle the problem!
Xian has 56 stamps and receives 12 new stamps per month.
After 1 month , Xian has 56 + 12 = 68 stamps
After 2 months , Xian has 56 + 12 × 2 = 80 stamps
After 3 months , Xian has 56 + 12 × 3 = 92 stamps
After n months , Xian has 56 + 12 × n = 56 + 12n stamps
Kai has 80 stamps and receives 8 new stamps per month.
After 1 month , Kai has 80 + 8 = 88 stamps
After 2 months , Kai has 80 + 8 × 2 = 96 stamps
After 3 months , Kai has 80 + 8 × 3 = 104 stamps
After n months , Kai has 80 + 8 × n = 80 + 8n stamps
If Xian and Kai will have the same number of stamps after n months , then :
Xian's stamps = Kai's stamps
56 + 12n = 80 + 8n
12n - 8n = 80 - 56
4n = 24
n = 24 / 4
n = 6
Xian and Kai will have the same number of stamps after 6 months.
Number of stamps that each of them has will be :
56 + 2n = 56 + 12(6) = 56 + 72 = 128
Grade: Middle School
Subject: Mathematics
Chapter: Arithmetic and Geometric Series
Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term
