SAT Consecutive Integer Problems: Setting Up and Solving Number Pattern Questions

Consecutive integers are integers that differ by 1 (e.g., n, n+1, n+2). Consecutive even integers differ by 2 and both must be even (e.g., n, n+2, n+4, where n is even). Consecutive odd integers also differ by 2 and both must be odd (e.g., n, n+2, n+4, where n is odd). For any word problem involving consecutive integers, let n represent the smallest and express each subsequent integer in terms of n. This converts the word problem into a simple algebraic equation.

Practice prompt: "Three consecutive integers sum to 48. Find them." Let the integers be n, n+1, n+2. Then n+(n+1)+(n+2)=48, so 3n+3=48, 3n=45, n=15. The integers are 15, 16, 17. Before writing any variable expression, confirm from the problem whether you are working with any integers, even integers, or odd integers, because this determines whether the common difference is 1 or 2.

SAT consecutive integer problems use three main conditions: (1) sum equals a given value; (2) product equals a given value; (3) the larger minus the smaller equals a given value. For difference conditions, the setup is direct: (n+1)-n=1 or (n+2)-n=2, so difference conditions often collapse to a one-step check rather than a full equation. For product conditions, set up n(n+1)=given value and use factoring or the quadratic formula to solve.

Micro-example: two consecutive integers have a product of 72. Set up n(n+1)=72. Expand: n^2+n-72=0. Factor: (n+8)(n-9)=0. Solutions: n=-8 or n=9. The positive pair is 9 and 10; the negative pair is -8 and -9 (both valid). When a product problem produces two solutions, check whether the question asks for positive integers, negative integers, or both, and only list the solutions that satisfy the given conditions.

Trap 1: setting up consecutive even integers as n, n+1, n+2 instead of n, n+2, n+4. Consecutive even integers always differ by 2, not 1. Trap 2: assuming the first integer must be positive. The problem may have negative solutions and both should be evaluated. Trap 3: confusing "sum of three consecutive integers equals X" with "each consecutive integer is X." Re-read the problem to confirm what quantity equals what value before writing an equation.

Error-prevention mini-routine: (1) identify integer type (any, even, odd); (2) assign n to the smallest; (3) write expressions for each integer using the correct common difference; (4) translate the condition into one equation; (5) solve and check all solutions against the original condition. After solving, substitute your answer back into the original word problem's condition, not just the equation, to catch setup errors that produce algebraically valid but contextually wrong answers.

Some SAT problems involve more than three consecutive integers or add a multiplier condition. Example: "Five consecutive even integers sum to 130." Let the integers be n, n+2, n+4, n+6, n+8. Their sum: 5n+20=130, so 5n=110, n=22. The integers are 22, 24, 26, 28, 30. Example with a multiplier: "One consecutive integer is three times another. Find both integers." Let the smaller be n and the larger be n+1. Then n+1=3n, so 2n=1, n=1/2. But 1/2 is not an integer, indicating this condition has no integer solution. Some SAT problems test whether you recognize when no solution exists.

Practice prompt: "Two consecutive odd integers have a sum of 10." Let n and n+2 be the integers. Then 2n+2=10, 2n=8, n=4. But 4 is even, so no odd solution exists. Always verify that your solution satisfies all constraints in the original problem, including the integer-type constraint, because algebraic answers that violate the integer-type requirement indicate the problem has no valid solution.