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Three-set counting is often a bookkeeping problem. Before using a formula, establish whether a pairwise overlap includes people who are also in the third set. That definition changes how the data should be interpreted.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 SETS 1018 and tell us your school year and question.

State what the overlap numbers include

In a fictional group of 60 students, 30 belong to club A, 28 to B and 20 to C. The pairwise overlaps are 12 in A and B, 10 in A and C, and 8 in B and C. Each pairwise number includes the five students in all three clubs.

These are invented counts for a teaching example. The sets represent membership, and each student is counted once in the overall group.

Count the union

Adding 30 + 28 + 20 counts students in two clubs twice and those in three clubs three times. Subtract the three pairwise overlaps, then add the triple overlap back once.

The union is 30 + 28 + 20 - 12 - 10 - 8 + 5 = 53. Therefore 60 - 53 = 7 students belong to none of the clubs.

Find membership in A only

Start with all 30 members of A. Subtract the 12 in A and B and the 10 in A and C. The five triple members have now been subtracted twice, so add them back once.

The number in A only is 30 - 12 - 10 + 5 = 13. This answer is different from the total number in exactly one club, which would also require the B-only and C-only regions.

Use the disjoint regions as a check

The pair-only regions are 7 in AB, 5 in AC and 3 in BC. The single-only regions are 13 in A, 13 in B and 7 in C. Together with five in all three, these sum to 53.

Starting with the central overlap and working outward is a reliable diagram method. All region counts should be nonnegative and should reconstruct each given total. This original problem supports AMC 12 preparation; current competition details should be checked through MAA.

Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.

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Planning a first attempt or reviewing a topic you find difficult? Scan the QR code and send AMC12 SETS 1018. Include your school year, country or competition region, experience and the question you want to discuss.

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Official information and further reading: Source 1

A recurrence can look almost geometric except for an added constant. A carefully chosen shift can remove that constant. The method is worth understanding because it explains the formula instead of guessing it from a short list of terms.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 SEQUENCES 1017 and tell us your school year and question.

Generate enough terms to check your work

Let a₁ = 1 and a_n+1 = 2a_n + 3 for integers n ≥ 1. The first five terms are 1, 5, 13, 29 and 61.

These terms are not a geometric sequence: the ratios are not constant. Repeated substitution will find a small-index term, but a transformation can reveal the general structure.

Choose a shift that absorbs the constant

Define b_n = a_n + 3. Then b_n+1 = a_n+1 + 3 = 2a_n + 6 = 2b_n.

The new sequence is geometric, with first term b₁ = 4 and common ratio 2. Hence b_n = 4 × 2^n-1, and a_n = 4 × 2^n-1 - 3.

Check the initial value and recurrence

At n = 1, the formula gives 4 - 3 = 1. At n = 5, it gives 64 - 3 = 61. Both agree with the direct calculation.

For a full structural check, substituting the formula into 2a_n + 3 gives 8 × 2^n-1 - 3, which is the formula for a_n+1. Checking only a few numerical terms would not establish that identity for every n.

Understand how the shift was found

If b_n = a_n + c, requiring b_n+1 = 2b_n gives 3 + c = 2c, so c = 3. The shift is chosen to make the algebra work.

Keep the starting index visible: a sequence beginning at n = 0 would use a different initial-value formula. This original exercise develops one algebra technique for AMC 12 preparation; it is not an official problem or a prediction of the paper.

Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.

Contact Hanlin

Planning a first attempt or reviewing a topic you find difficult? Scan the QR code and send AMC12 SEQUENCES 1017. Include your school year, country or competition region, experience and the question you want to discuss.

Contact Hanlin on WeChat

On your phone, use the button or save the QR image and scan it within WeChat.

Official information and further reading: Source 1