If the list of letters, P, R, S, T, U is an arithmetic sequence, which of the…

Key idea: Multiplying all terms by the same constant or adding/subtracting the same constant preserves the common difference; squaring does not generally preserve it.

• Write the terms of the arithmetic sequence as P, P + d, P + 2d, P + 3d, P + 4d, where d is the common difference.

• Multiplying each term by 2 gives 2P, 2(P + d), 2(P + 2d), … The difference between consecutive terms is 2d, so these terms form an arithmetic sequence.

• Subtracting 3 from each term gives P − 3, (P + d) − 3, (P + 2d) − 3, … The differences remain d, so these terms also form an arithmetic sequence.

• Squaring the terms gives (P + kd)^2 for k = 0, 1, 2, 3, 4. The difference between consecutive squared terms equals

• (P + (k + 1)d)^2 − (P + kd)^2 = 2Pd + (2k + 1)d^2, which depends on k unless d = 0. Therefore the differences are not constant in general.

• Concrete counterexample: choose P = 1 and d = 2 so the arithmetic sequence is 1, 3, 5, 7, 9. Squaring gives 1, 9, 25, 49, 81 with consecutive differences 8, 16, 24, 32, which are not constant.

Conclusion: Multiplying all terms by 2 and subtracting 3 both preserve an arithmetic sequence, but squaring does not in general. Therefore the transformations that multiply all terms by 2 and subtract 3 produce arithmetic sequences.