Question

What kind of sequence is this? 206, 190, 174, 158

Answer 1

Arithmetic

Explanation:

Arithmetic Sequence is a sequence that has common difference - a common difference is same difference in a sequence. If we take next term to subtract the previous term and you keep getting same difference or value then that's a common difference. It can be mathematically expressed as an equation of:

`\displaystyle{d=a_(n+1)-a_n}`

Now, I'll show why the sequence is arithmetic - take the sequence to subtract for common difference:

• 190 - 206 = -16
• 174 - 190 = -16
• 158 - 174 = -16

Notice how we keep getting -16 after subtracting previous term with next term. Hence this sequence is arithmetic sequence for sure!

Geometric Sequence is a sequence with common ratio - a common ratio is similar to common difference except ratio is all about division. Since common difference is expressed as subtraction of next term and previous term then common ratio is division of next term and previous term which can be mathematically expressed as:

`\displaystyle{r = (a_(n+1))/(a_n)}`

However this sequence is not geometric sequence because if you divide next term with previous term then you'll keep getting difference result and common ratio means having same ratio or constant ratio for entire sequence.

Now see why this sequence cannot be geometric sequence:

• 190/206 = 0.922
• 174/190 = 0.915
• 158/174 = 0.908

Notice that none of values are equal and thus, this sequence doesn't have common ratio which leads to not being geometric sequence.