Jennifer B. answered • 10/08/20
Experienced teacher and tutor specializing in math and physics.
A formula is a pattern between values. Geometric sequences will have a constant ratio for consecutive values. This ratio will give you the multiplier for the next value given a previous one.
Ask yourself: what do I multiply the first value by to get the second?
-6(x)= -24 solving for x: x = (-24)/(-6) this is our ratio or multiplier for consecutive values. In this case, x = 4. You can check other consecutive values to see if their ratio is the same. If not, it is not a geometric sequence. If so, then it is (assuming the calculation was done correctly).
Look for a pattern:
Using our ratio: -6(4) = -24; -24 (4) = -96; -96(4) = ? Note that we are repeatedly multiplying by 4. Repeated multiplication is represented with exponents: 42 = 4(4); 43 = 4(4)(4); 427 = (27 fours multiplied together).... Thus, we can write a formula to determine any value in the sequence as long as we know the first value: an = a1(r)n-1 where n is the position in the sequence.
For the given sequence, a1 = -6, r = 4. Thus, our formula would be: an = -6(4)n-1
Now we can find any term in the sequence!
an = -6(4)n-1
Want the 4th term in the sequence? a4 = -6(4)4-1 ; a4 = -6(4)3 = -6(4)(4)(4) =
27th term? a27 = -6(4)27-1 = -6(4)26
5th term? just replace then n's with 5.
*******NOTE: often the formula will be written: an = -6(4)n where n=0 is the first term. This is the exact same formula, it just looks different. Using this one, the first term is n = 0, the second n = 1, the 27th n = 26. That is why I write it the way I did. Then the first term is n = 1, the 27th is n = 27, and so on. The only difference is what we call the first term: a0 or a1 . *******
Joey B.
thank you10/08/20