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"Three numbers in AP have sum 24 and product 440. Find them." If you write the numbers as a, a + d, a + 2d, you get two equations in two unknowns and a cubic to untangle. If you write them as a − d, a, a + d, the sum gives a = 8 in one line. This post shows how to choose the terms for APs and GPs, where the four-term versions go wrong, and works a mixed AP-GP problem that has two valid answers.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus lists "Arithmetic and Geometric progressions" under sequences and series, and the JEE (Advanced) 2026 brochure lists arithmetic and geometric progressions and their sums. In the current NCERT books, APs are taught in the Class 10 chapter Arithmetic Progressions, and the Class 11 chapter Sequences and Series (2026-27 reprint) covers GPs, the arithmetic and geometric means and the relation between them. Neither chapter teaches the symmetric choice of terms as a method, but the Class 11 exercises use it, for example in a question on three numbers in GP with sum 56. Check the current documents for your exam.
Why choose symmetric terms?
Because the condition in the question decides which choice cancels.
AP with a given sum: a − d, a, a + d add to 3a.
GP with a given product: a/r, a, ar multiply to a³.
One unknown disappears immediately, and the other comes from the second condition.
Worked example 1: three numbers in AP
Three numbers in AP have sum 24 and product 440.
Write them as a − d, a, a + d. Sum: 3a = 24, so a = 8.
Product: (8 − d) × 8 × (8 + d) = 8(64 − d²) = 440, so 64 − d² = 55 and d² = 9.
d = 3 or d = −3. Both give the same three numbers, 5, 8, 11, in opposite orders.
Check: 5 + 8 + 11 = 24 and 5 × 8 × 11 = 440.
Worked example 2: three numbers in GP
Three numbers in GP have product 216 and sum 19.
Write them as a/r, a, ar. Product: a³ = 216, so a = 6.
Sum: 6/r + 6 + 6r = 19. Multiply by r: 6r² − 13r + 6 = 0.
r = (13 ± 5)/12, so r = 3/2 or r = 2/3.
Both give 4, 6, 9, in one order or the other.
Check: 4 × 6 × 9 = 216 and 4 + 6 + 9 = 19.
What changes with four terms?
For four numbers in AP, the symmetric choice is a − 3d, a − d, a + d, a + 3d. The common difference is 2d, not d. Forgetting this is the most common slip.
Worked example 3. Four numbers in AP have sum 20 and the sum of their squares is 120.
Sum: 4a = 20, so a = 5.
Squares: (a − 3d)² + (a − d)² + (a + d)² + (a + 3d)² = 4a² + 20d², because the cross terms cancel. So 100 + 20d² = 120 and d² = 1.
With d = 1, the numbers are 2, 4, 6, 8. The common difference is 2, which is 2d.
Check: 4 + 16 + 36 + 64 = 120.
For four numbers in GP, the matching choice is a/r³, a/r, ar, ar³, with common ratio r². A real r² is never negative, so this choice cannot produce a GP such as 1, −2, 4, −8, whose ratio is −2. If the ratio might be negative, write a, ar, ar², ar³ instead.
Worked example 4: an AP that becomes a GP
Three numbers in AP have sum 15. If 1, 4 and 19 are added to them in order, the results are in GP. Find the numbers.
Write them as 5 − d, 5, 5 + d, since 3a = 15 gives a = 5.
After adding: 6 − d, 9, 24 + d.
For a GP, the middle term squared equals the product of the outer terms: 81 = (6 − d)(24 + d).
Expand: 144 − 18d − d² = 81, so d² + 18d − 63 = 0, giving (d − 3)(d + 21) = 0.
d = 3 gives 2, 5, 8, which become 3, 9, 27, a GP with ratio 3.
d = −21 gives 26, 5, −16, which become 27, 9, 3, a GP with ratio 1/3.
Both answers satisfy every condition. A student who stops at d = 3 loses marks on a question that asks for all possible sets.
Which mistakes cost marks?
Using a, a + d, a + 2d when the sum is given, and drowning in algebra.
Taking d as the common difference in a − 3d, a − d, a + d, a + 3d.
Using a/r³, a/r, ar, ar³ when the ratio could be negative.
Dropping the second root of a quadratic in d or r.
Not checking the final numbers against both conditions.
A practice routine
Do NCERT Class 11 Miscellaneous Exercise question 4, the GP with sum 56, using a/r, a, ar.
Make up one AP and one GP problem of your own, solve each, and swap with a friend.
Rework example 4 with "adds 1, 4 and 19" changed to "adds 1, 3 and 9", and check how many answers you get.
Revise the basics in our Class 10 arithmetic progressions guide, and see an infinite GP at work in our post on recurring decimals as fractions.
If you can see which terms to choose but the algebra keeps going wrong, the Fermi Problem Library has a step-by-step solver that explains the reasoning at each line.
Today, solve example 1 again with a, a + d, a + 2d, and time both methods.
