Definition: The common ratio is the constant multiplier that takes each term of a geometric sequence to the next term; it equals any term divided by the term before it.
What is it?
A geometric sequence grows or shrinks by multiplication. The multiplier is the common ratio, written r. If |r| is greater than 1, the terms get larger in size. If r is between -1 and 1, they shrink toward zero.
How do you find r?
Divide a term by the previous one: r = an+1 / an. Confirm with a second pair.
Example. In 1200, 300, 75, 18.75, we get 300 / 1200 = 0.25 and 75 / 300 = 0.25, so r = 0.25. If you know a2 = 12 and a5 = 324, then three multiplications connect them, so r3 = 324 / 12 = 27 and r = 3.
What mistakes occur?
Learners subtract instead of divide, which finds a difference, not a ratio. Others divide the later term into the earlier one and get the reciprocal. Also remember that a negative ratio flips the sign each step, so the terms alternate between positive and negative. The ratio is the growth factor of the matching exponential function.