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When a question asks for a symmetric expression, finding each possible value of the variable can introduce radicals that later cancel. An identity may determine the requested expression directly from the information given.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 RECIPROCAL SUM 1031 and tell us your school year and question.

Keep the nonzero condition visible

In this original exercise, x is real and nonzero, and x + 1/x = 3. Find x³ + 1/x³.

Solving x² - 3x + 1 = 0 is possible, but the requested sum treats x and its reciprocal symmetrically. First ask whether the given sum can be transformed into the target.

Square with the cross term included

Squaring the given equation gives x² + 2 + 1/x² = 9, because x(1/x) = 1. Therefore x² + 1/x² = 7.

The answer is not nine: that would omit the cross term. Stating why the product is one also identifies precisely where the nonzero condition is used.

Multiply and collect the middle terms

Multiply the quadratic sum by the original sum. The left side expands to x³ + x + 1/x + 1/x³, while the right side is 7 × 3 = 21.

The two middle terms add to three. Subtracting them leaves x³ + 1/x³ = 18. No individual root has been selected, so the result applies to every x satisfying the original condition.

Generalise only after checking the first steps

If S_n = xⁿ + x^-n, then S₀ = 2 and S₁ = 3. Multiplication gives S_n+1 = 3S_n - S_n-1 for integers n ≥ 1. Thus S₂ = 7 and S₃ = 18 agree with the direct work.

This is an original AMC 12 practice problem, not an official question. The habit transfers to more extended algebra practice, including preparation for AIME: identify the target structure, justify the identity and check conditions before doing extra algebra.

Continue reading: AMC 12 Sequences: Derive a Geometric Sum by Subtracting Two Lines.

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Official information and further reading: Source 1

Evaluating a polynomial at 1 adds its coefficients. Evaluating at -1 changes the signs of its odd-power terms. Combining those two evaluations can isolate a whole group of coefficients at once.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 EVEN COEFFICIENTS 1030 and tell us your school year and question.

Define the two sums

In this original example, let E be the sum of C(7,k) over even k, and O the corresponding sum over odd k. The binomial expansion of (1 + x)⁷ contains exactly these coefficients.

Substituting x = 1 gives E + O = 2⁷ = 128. Substituting x = -1 gives E - O = 0. The second equality uses the signs of the powers, not the signs of the positive coefficients themselves.

Combine the equations

Adding the two equations gives 2E = 128, so E = 64. Subtracting them gives O = 64 as well.

For a direct numerical check, the even-index sum is C(7,0) + C(7,2) + C(7,4) + C(7,6) = 1 + 21 + 35 + 7 = 64. The evaluation method obtains that total without separately computing the four coefficients.

Explain the equality by pairing subsets

Take a set of seven labelled objects and choose one distinguished object. Pair every subset not containing it with the subset obtained by adding it. Each pair has one even-sized subset and one odd-sized subset.

Every subset appears in exactly one pair. Since there are 128 subsets altogether, each parity class has 64. This is an independent counting explanation of the same identity.

State the range of the general result

For a positive integer n, the same reasoning gives equal even- and odd-index binomial sums, each 2^n-1. The empty-set case n = 0 needs separate treatment: its only subset is even-sized, so the two totals are not equal.

This is an original AMC 12 learning exercise, not an official problem. When using evaluations to filter terms, state which terms keep or reverse their signs and why the resulting equations determine the requested sum.

Continue reading: AMC 12 Algebra: Find One Binomial Coefficient Without Expanding Everything.

Contact Hanlin

Planning a first attempt or reviewing a topic you find difficult? Scan the QR code and send AMC12 EVEN COEFFICIENTS 1030. 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