IB AI · Unit 2 · 2A Arithmetic Sequences
Three consecutive terms share one common difference. Set the two gaps equal and solve.
Jump to section:
Equal gaps
Three terms are consecutive in an arithmetic sequence when the jump from the first to the second equals the jump from the second to the third.
\((second) - (first) = (third) - (second)\)
Solve for the variable
Expand each side, then move the variable to one side and the numbers to the other side.
The common difference is positive.
The gap from the first term to the second equals the gap from the second to the third.
\(\left(5x-4\right) - \left(2x+1\right) = \left(3x+6\right) - \left(5x-4\right)\)
\(3x-5 = -2x+10\)
\(3x - \left(-2x\right) = 10 - \left(-5\right)\)
\(5x = 15\)
\(x = \dfrac{15}{5}\)
\(x = 3\)
Answer: \(x = 3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(4x+2\right) - \left(x+3\right) = \left(2x+11\right) - \left(4x+2\right)\)
\(3x-1 = -2x+9\)
\(3x - \left(-2x\right) = 9 - \left(-1\right)\)
\(5x = 10\)
\(x = \dfrac{10}{5}\)
\(x = 2\)
Answer: \(x = 2\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(x+8\right) - \left(3x-2\right) = \left(4x-2\right) - \left(x+8\right)\)
\(-2x+10 = 3x-10\)
\(-2x - \left(3x\right) = -10 - \left(10\right)\)
\(-5x = -20\)
\(x = \dfrac{-20}{-5}\)
\(x = 4\)
Answer: \(x = 4\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(x+9\right) - \left(4x-3\right) = \left(3x+11\right) - \left(x+9\right)\)
\(-3x+12 = 2x+2\)
\(-3x - \left(2x\right) = 2 - \left(12\right)\)
\(-5x = -10\)
\(x = \dfrac{-10}{-5}\)
\(x = 2\)
Answer: \(x = 2\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(5x-4\right) - \left(2x+2\right) = \left(x+11\right) - \left(5x-4\right)\)
\(3x-6 = -4x+15\)
\(3x - \left(-4x\right) = 15 - \left(-6\right)\)
\(7x = 21\)
\(x = \dfrac{21}{7}\)
\(x = 3\)
Answer: \(x = 3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(5x-9\right) - \left(x+2\right) = \left(2x+8\right) - \left(5x-9\right)\)
\(4x-11 = -3x+17\)
\(4x - \left(-3x\right) = 17 - \left(-11\right)\)
\(7x = 28\)
\(x = \dfrac{28}{7}\)
\(x = 4\)
Answer: \(x = 4\)
The sequence decreases, so the common difference is negative.
The gap from the first term to the second equals the gap from the second to the third.
\(\left(x+4\right) - \left(3x+2\right) = \left(4x-9\right) - \left(x+4\right)\)
\(-2x+2 = 3x-13\)
\(-2x - \left(3x\right) = -13 - \left(2\right)\)
\(-5x = -15\)
\(x = \dfrac{-15}{-5}\)
\(x = 3\)
Answer: \(x = 3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(x+4\right) - \left(4x+1\right) = \left(5x-7\right) - \left(x+4\right)\)
\(-3x+3 = 4x-11\)
\(-3x - \left(4x\right) = -11 - \left(3\right)\)
\(-7x = -14\)
\(x = \dfrac{-14}{-7}\)
\(x = 2\)
Answer: \(x = 2\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(3x-6\right) - \left(x+7\right) = \left(2x-7\right) - \left(3x-6\right)\)
\(2x-13 = -x-1\)
\(2x - \left(-x\right) = -1 - \left(-13\right)\)
\(3x = 12\)
\(x = \dfrac{12}{3}\)
\(x = 4\)
Answer: \(x = 4\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(2x+3\right) - \left(5x-4\right) = \left(4x-5\right) - \left(2x+3\right)\)
\(-3x+7 = 2x-8\)
\(-3x - \left(2x\right) = -8 - \left(7\right)\)
\(-5x = -15\)
\(x = \dfrac{-15}{-5}\)
\(x = 3\)
Answer: \(x = 3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(6x-7\right) - \left(3x+4\right) = \left(2x-4\right) - \left(6x-7\right)\)
\(3x-11 = -4x+3\)
\(3x - \left(-4x\right) = 3 - \left(-11\right)\)
\(7x = 14\)
\(x = \dfrac{14}{7}\)
\(x = 2\)
Answer: \(x = 2\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(4x-11\right) - \left(x+8\right) = \left(2x-5\right) - \left(4x-11\right)\)
\(3x-19 = -2x+6\)
\(3x - \left(-2x\right) = 6 - \left(-19\right)\)
\(5x = 25\)
\(x = \dfrac{25}{5}\)
\(x = 5\)
Answer: \(x = 5\)
Find x, then write the three terms and the common difference.
The gap from the first term to the second equals the gap from the second to the third.
\(\left(-x+10\right) - \left(2x+3\right) = \left(4x-7\right) - \left(-x+10\right)\)
\(-3x+7 = 5x-17\)
\(-3x - \left(5x\right) = -17 - \left(7\right)\)
\(-8x = -24\)
\(x = \dfrac{-24}{-8}\)
\(x = 3\)
Substitute that value back into the three terms.
\(2x+3 = 2\left(3\right)+3\)
\(2\left(3\right)+3 = 6+3\)
\(6+3 = 9\)
\(-x+10 = -3+10\)
\(-3+10 = 7\)
\(4x-7 = 4\left(3\right)-7\)
\(4\left(3\right)-7 = 12-7\)
\(12-7 = 5\)
\(d = 7 - 9\)
\(d = -2\)
Answer: \(x = 3\), the terms are 9, 7, and 5, and the common difference = \(-2\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(-x+14\right) - \left(2x+1\right) = \left(3x+6\right) - \left(-x+14\right)\)
\(-3x+13 = 4x-8\)
\(-3x - \left(4x\right) = -8 - \left(13\right)\)
\(-7x = -21\)
\(x = \dfrac{-21}{-7}\)
\(x = 3\)
Substitute that value back into the three terms.
\(2x+1 = 2\left(3\right)+1\)
\(2\left(3\right)+1 = 6+1\)
\(6+1 = 7\)
\(-x+14 = -3+14\)
\(-3+14 = 11\)
\(3x+6 = 3\left(3\right)+6\)
\(3\left(3\right)+6 = 9+6\)
\(9+6 = 15\)
\(d = 11 - 7\)
\(d = 4\)
Answer: \(x = 3\), the terms are 7, 11, and 15, and the common difference = \(4\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(3x+1\right) - \left(-2x+8\right) = \left(x+8\right) - \left(3x+1\right)\)
\(5x-7 = -2x+7\)
\(5x - \left(-2x\right) = 7 - \left(-7\right)\)
\(7x = 14\)
\(x = \dfrac{14}{7}\)
\(x = 2\)
Substitute that value back into the three terms.
\(-2x+8 = -2\left(2\right)+8\)
\(-2\left(2\right)+8 = -4+8\)
\(-4+8 = 4\)
\(3x+1 = 3\left(2\right)+1\)
\(3\left(2\right)+1 = 6+1\)
\(6+1 = 7\)
\(x+8 = 2+8\)
\(2+8 = 10\)
\(d = 7 - 4\)
\(d = 3\)
Answer: \(x = 2\), the terms are 4, 7, and 10, and the common difference = \(3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(-2x+15\right) - \left(3x-2\right) = \left(2x-4\right) - \left(-2x+15\right)\)
\(-5x+17 = 4x-19\)
\(-5x - \left(4x\right) = -19 - \left(17\right)\)
\(-9x = -36\)
\(x = \dfrac{-36}{-9}\)
\(x = 4\)
Substitute that value back into the three terms.
\(3x-2 = 3\left(4\right)-2\)
\(3\left(4\right)-2 = 12-2\)
\(12-2 = 10\)
\(-2x+15 = -2\left(4\right)+15\)
\(-2\left(4\right)+15 = -8+15\)
\(-8+15 = 7\)
\(2x-4 = 2\left(4\right)-4\)
\(2\left(4\right)-4 = 8-4\)
\(8-4 = 4\)
\(d = 7 - 10\)
\(d = -3\)
Answer: \(x = 4\), the terms are 10, 7, and 4, and the common difference = \(-3\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(-3x+11\right) - \left(4x-1\right) = \left(2x+11\right) - \left(-3x+11\right)\)
\(-7x+12 = 5x\)
\(-7x - \left(5x\right) = 0 - \left(12\right)\)
\(-12x = -12\)
\(x = \dfrac{-12}{-12}\)
\(x = 1\)
Substitute that value back into the three terms.
\(4x-1 = 4\left(1\right)-1\)
\(4\left(1\right)-1 = 4-1\)
\(4-1 = 3\)
\(-3x+11 = -3\left(1\right)+11\)
\(-3\left(1\right)+11 = -3+11\)
\(-3+11 = 8\)
\(2x+11 = 2\left(1\right)+11\)
\(2\left(1\right)+11 = 2+11\)
\(2+11 = 13\)
\(d = 8 - 3\)
\(d = 5\)
Answer: \(x = 1\), the terms are 3, 8, and 13, and the common difference = \(5\)
The gap from the first term to the second equals the gap from the second to the third.
\(\left(2x-4\right) - \left(-x+9\right) = \left(4x-12\right) - \left(2x-4\right)\)
\(3x-13 = 2x-8\)
\(3x - \left(2x\right) = -8 - \left(-13\right)\)
\(x = 5\)
\(x = 5\)
Substitute that value back into the three terms.
\(-x+9 = -5+9\)
\(-5+9 = 4\)
\(2x-4 = 2\left(5\right)-4\)
\(2\left(5\right)-4 = 10-4\)
\(10-4 = 6\)
\(4x-12 = 4\left(5\right)-12\)
\(4\left(5\right)-12 = 20-12\)
\(20-12 = 8\)
\(d = 6 - 4\)
\(d = 2\)
Answer: \(x = 5\), the terms are 4, 6, and 8, and the common difference = \(2\)
Need help with consecutive terms?
Book a Session with Amy