KCSE 2025 Maths P2 Q1 — Arithmetic Progression, Finding x and the Common Difference
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The Question
“The first three terms of an arithmetic progression (AP) are (8 minus x), 2x and (3x plus 2). Find the value of x and hence the common difference of the AP.”
Use the defining property of an AP
In an arithmetic progression the gap between any term and the one before it is always the same value, called the common difference. So the second term minus the first term must equal the third term minus the second term. Setting these two gaps equal gives you a single equation in x.
Simplify each side
On the left, subtracting the bracket flips the signs inside it, so minus the quantity (8 minus x) becomes minus 8 plus x. Combine the x terms. On the right, the two x terms partly cancel, leaving a much simpler expression.
Collect like terms and solve for x
Bring the x terms to one side and the numbers to the other. Take x from both sides and add 8 to both sides, then divide by the coefficient of x to isolate it.
Substitute back and find the common difference
Put x = 5 into each of the three terms to get the actual numbers in the progression. The common difference is then the second term minus the first term. Confirm it with the third term minus the second.
Final Result
x = 5. The three terms are 3, 10 and 17, so the common difference is 7.
Why this method works
An arithmetic progression is defined by adding the same fixed amount to move from one term to the next. That means the difference between consecutive terms is constant, so equating the first gap (second minus first) with the second gap (third minus second) captures the whole structure of the sequence in one equation. Solving it pins down x, and once the terms are known numerically the common difference is just any consecutive difference.
With x = 5 the terms are 3, 10, 17. The gap 10 - 3 = 7 equals the gap 17 - 10 = 7, so the common difference is consistent.