I keep tripping over geometric sequences and could use a sanity check. I get that it’s the “multiply by the same number each time” idea, like tapping the same button on a calculator over and over. But when I’m given two terms that aren’t next to each other, I’m not sure how to pin down the common ratio and write the nth-term formula without mixing it up with arithmetic sequences.
Here’s a simple example I was trying: suppose a₂ = 12 and a₅ = 96. My first (wrong) instinct was to look at the difference: 96 − 12 = 84, then divide by 3 steps to get 28… but I know that’s arithmetic thinking, not geometric. Then I tried 96 ÷ 12 = 8, and since that’s from term 2 to term 5 (three jumps), I figured maybe r^3 = 8, so r = 2? If that’s right, does that mean a₁ would be 6 and the formula is something like a_n = something × 2^(n−1)? I always second-guess the (n−1) part.
Where I get extra confused is with signs and fractions. For example, if the sequence looks like 8, −4, 2, −1, 0.5, … do I just take r = −1/2 and write the nth term normally, or is there a better way to handle the alternating signs? And if one of the given terms is 0, is that automatically not geometric (unless everything is 0)?
Could someone explain a reliable way to go from two non-adjacent terms to the common ratio and the nth-term formula, and maybe point out what I’m doing right/wrong in my attempt above?
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3 Responses
Love this! The reliable recipe is: if you know two non-adjacent terms a_k and a_m with m−k = d, then r^d = a_m / a_k, so r = (a_m / a_k)^(1/d) (with the caveat that if d is odd, r can be negative; if d is even, a_m/a_k must be positive to get a real r). Then any term is a_n = a_k · r^(n−k), and in particular a_n = a_1 · r^(n−1). For your example, a₂ = 12 and a₅ = 96 gives r^3 = 96/12 = 8, so r = 2; then a₁ = a₂ / r = 6, and the nth term is a_n = 6 · 2^(n−1) (yes, that n−1 is exactly the “number of jumps” from the first term). For the alternating sequence 8, −4, 2, −1, 0.5, … the common ratio really is r = −1/2, so a_n = 8 · (−1/2)^(n−1); equivalently, a_n = 8 · (−1)^(n−1) · 2^(1−n). About zeros: if any term is 0, then either a_1 = 0 (so everything is 0), or r = 0 (so from a₂ onward everything is 0); there isn’t a nontrivial geometric sequence with a single isolated zero. My personal “aha!” moment was when I stopped thinking in differences and started counting jumps: the exponent is exactly how many times you “tap the ratio button” to get from one term to another-once I started writing r^(m−k) = a_m/a_k in the margin, the fog cleared and the signs/fractions became a fun puzzle rather than a trap.
A geometric sequence has the form a_n = a_1 · r^(n−1). If you know two non-adjacent terms a_i and a_j, divide to get r^(j−i) = a_j / a_i, so r = (a_j / a_i)^(1/(j−i)), and then a_1 = a_i / r^(i−1). One small wrinkle: if j−i is odd and a_j/a_i > 0, both r and −r fit that single equation; you need one more term to decide the sign. If a_j/a_i < 0 and j−i is even, there is no real r (you’d need complex numbers). A concise reference: Khan Academy’s geometric sequences intro: https://www.khanacademy.org/math/algebra/sequences/geometric-sequences/a/intro-to-geometric-sequences-with-variable-r
Worked example (yours): a_2 = 12 and a_5 = 96. Then r^3 = 96/12 = 8, so r = 2. Hence a_1 = a_2 / r = 12/2 = 6, and a_n = 6 · 2^(n−1). Quick check: a_5 = 6 · 2^4 = 96, as required. For alternating signs like 8, −4, 2, −1, 0.5, … the ratio is r = −1/2, so a_n = 8 · (−1/2)^(n−1); the alternating signs are handled naturally by the negative ratio.
About zeros: if some term is zero and r ≠ 0, then a_1 must be zero, which forces every term to be zero. If r = 0, the sequence is a_1, 0, 0, 0, …; that’s the only geometric case mixing a zero term with a nonzero term.
You’ve got it: if you know a_j and a_k, use r = (a_k/a_j)^(1/(k−j)) and then a_n = a_j·r^(n−j); for your example a2=12 and a5=96 give r^3=8 so r=2, a1=6, and a_n=6·2^(n−1) (the n−1 makes a1 use r^0=1). Alternating signs and fractions handle themselves-8, −4, 2, −1, 0.5 has r=−1/2 so a_n=8(−1/2)^(n−1)-and if any term is 0 then all must be 0 (otherwise not geometric); I remember scribbling “differences” in the margins until this clicked, and this quick primer helped me a lot: https://www.khanacademy.org/math/algebra/sequences/alg1-geometric-sequences/a/geometric-sequences-intro.