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3 Responses
I mix this up all the time too, so you’re not alone! The “add (n−1)r” formula is for arithmetic sequences (constant difference), not geometric ones. For geometric sequences you multiply by the common ratio each step, and the formula is a_n = a1 · r^(n−1). Here a1 = 5 and r = 2, so you double each time: 5, 10, 20, 40. As a simple check, a4 = 5 · 2^(4−1) = 5 · 2^3 = 5 · 6 = 30-oops, my brain did a somersault there-2^3 is 8, so it’s actually 5 · 8 = 40. Thinking “aren’t you just supposed to add the ratio?” kind of makes sense in the sense that each step “adds one more factor of 2” in the exponent, but if you literally add 2 each time you’re treating it like an arithmetic sequence, which is why you landed on 11.
You used the arithmetic formula; in a geometric sequence you multiply by the common ratio each step, so a_n = 5·2^(n−1) and a_4 = 5·2^3 = 40.
Does it help to remember the rule of thumb: add a common difference → arithmetic, multiply a common ratio → geometric?
Great instinct finding r = 2! The hiccup is mixing up the two big families of sequences. Arithmetic sequences add the same amount each step (like paying a flat $5 every week), so they use a_n = a1 + (n−1)d. But 5, 10, 20 is geometric: each term is multiplied by the same ratio (like doubling your stack of pancakes every flip), so the formula is a_n = a1 · r^(n−1). Here, a4 = 5 · 2^(4−1) = 5 · 8 = 40. That’s why adding 2 gave 11-it’s the arithmetic formula with the wrong model.
A good quick check: differences aren’t constant here (10−5 = 5, 20−10 = 10), but ratios are (10/5 = 2, 20/10 = 2). Geometric = multiply by r; arithmetic = add d. If you want a clear refresher, this Khan Academy intro to geometric sequences walks through exactly this: https://www.khanacademy.org/math/algebra/sequences/alg1-introduction-to-geometric-sequences/v/introduction-to-geometric-sequences
Follow-up: Want to try another? For the sequence 3, 6, 12, what’s the general formula and what is a5? Or, how would you very quickly decide if 7, 14, 28, 56 is arithmetic or geometric?