# Problem Solving What is the teacher response to Joey saying that 4,24,44, and 64 all have remainder 0 when divided by 4 , so all numbers that end in 4 must have 0 remainder when divided by 4. Al...

Problem Solving

What is the teacher response to Joey saying that 4,24,44, and 64 all have remainder 0 when divided by 4 , so all numbers that end in 4 must have 0 remainder when divided by 4.

Al and Betty were asked to extend the sequence 2,4,8.... Al said his answer 2,4,8,16,32,64,... was the correct one. Betty said Al was wrong and it should be 2.4,8,14,22,32,44,... What do a teacher say to the students?

A student claims the sequence 6,6,6,6,6,...never changes, so it is neither arithmetic nor geometric. How do a teacher respond? a student claims the sequence 6ˏ6ˏ6ˏ6ˏ6ˏ never changesˏ so it is neither arithmetics nor geometric . How do you respond

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Joey’s conclusion that “All numbers that end in 4 must have 0 remainder when divided by 4” **is not true.** Numbers like 14, 34, 54, 94, 134 etc. all end in 4 yet they do not have 0 as remainder when divided by 4. Let us consider the numbers 144 and 1234. The last two digits of the respective numbers are 44 and 34. 44 is divisible by 4 and so do 144. On the other hand 34 is not divisible by 4, and 1234 when divided by 4 leaves a remainder of 2. So it’s worth saying that “If any number ends in a two digit number that is divisible by 4 then the whole number will be divisible by 4 regardless of what is before the last two digits.”

The sequence 2, 4, 8….. represents a **geometric sequence** where each new number is obtained by multiplying the previous number by a selected number. This selected number is called the common ratio. Here, the common ratio is 4/2=8/4=2. Hence, the immediate number after 8 in the series would be 8*2=16, next 16*2=32 and so on. Finally, the series would be:

2, 4, 8, 16, 32, 64………

Again, the sequence 2, 4, 8, 14, 22, 32, 44 ... represents a **quadratic sequence** where the difference between the two consecutive numbers are in arithmetic progression.

2, 4, 8, 14, 22, 32, 44 ...

A difference line can be derived as: (2,4,6,8,10,12...), which would be in A.P.. **Therefore, both Al and Betty are correct.**

The series 6,6,6,6,6…… **is both an arithmetic and a geometric series** because the common difference between the consecutive numbers in the series is (6-6)=0 and the common ratio is 6/6=1.