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3 Responses
In a geometric sequence each term is obtained by multiplying the previous one by a constant ratio. Here, the ratio is r = 15/5 = 3 (and 45/15 = 3 as a check). So the next term is 45 × 3 = 135, not 60. Thinking in differences leads to arithmetic sequences; for geometric ones, always compare terms by division.
The nth-term formula for a geometric sequence is a_n = a_1 × r^(n−1). With a_1 = 5 and r = 3, we get a_n = 5 × 3^(n−1). Worked example: for n = 4, a_4 = 5 × 3^3 = 5 × 27 = 135, matching the next term we found. A short refresher is here: https://www.khanacademy.org/math/algebra/sequences/alg1-intro-to-geometric-sequences/v/geometric-sequences-introduction
You’re right to catch that you were adding-geometric sequences multiply by a constant ratio r each step. To find r, divide a term by the previous one: r = 15/5 = 3, and a quick check 45/15 = 3 confirms it. So each term is triple the last, meaning the next term is 45 × 3 = 135. The general nth-term formula for a geometric sequence with first term a1 and ratio r is a_n = a1 · r^(n−1), so here a_n = 5 · 3^(n−1). Quick verification: a4 = 5 · 3^3 = 5 · 27 = 135, which matches. Simple worked example: for 2, 6, 18, … we have r = 3, so a_n = 2 · 3^(n−1), and the fourth term is 2 · 3^3 = 54. If you want a tidy refresher on the difference between arithmetic and geometric sequences and how to get these formulas, this Khan Academy page is clear: https://www.khanacademy.org/math/algebra/sequences/alg1-intro-to-geometric-sequences/a/geometric-sequences-intro
I’ve totally done the “add 15 then add 30” thing too, so you’re in good company! For a geometric sequence we multiply by the same number each time-the common ratio-so from 5 to 15 is ×3 and from 15 to 45 is also ×3, meaning r = 3 (you can check by 15/5 = 3 and 45/15 = 3). That makes the next term 45 × 3 = 135, not 60-my brain still wants to sneak arithmetic in there sometimes. The general nth-term formula is a_n = a_1 r^(n−1), so here a_n = 5 · 3^(n−1). I always mix up the n vs n−1 bit and am tempted to write 5 · 3^n, which would accidentally make the first term 15 instead of 5 (unless you start counting at n = 0, which I sometimes do by mistake). Quick check: plugging n = 4 gives a_4 = 5 · 3^3 = 135. Simple worked example with a different sequence: for 2, 6, 18, ?, r = 3 and I’d say a_n = 2 · 3^(n−1); so the 4th term would be 2 · 3^4 = 162 (hmm, I might be overthinking the exponent again there). If you want a clear refresher, Khan Academy’s intro to geometric sequences is great: https://www.khanacademy.org/math/algebra/sequences/alg-geom-seq/a/geometric-sequences-intro