telescoping-collapse
A long sum or product is evaluated without ever computing an interior term: every term is rewritten in a paired form — f(n) minus f(n+1) for a sum, or f(n+1) over f(n) for a product — so consecutive pieces annihilate and only the two boundary values survive. The decisive move is finding the rewrite that manufactures the cancellation; evaluating the pieces independently is exactly the wrong instinct.
풀이 전략
- Keep the (1 + 1/n) product shape but move the upper limit so the collapse lands on a different clean integer
- Raise shift_distance from 1 to 2 (e.g. 1/(k(k+2))) so four boundary terms survive instead of two — same strategy, one grade harder
- Swap rewrite_identity from a plain fraction to a logarithm or a factorial while keeping the collapse skeleton identical
세부 유형 분포 (3)
한 줄을 클릭하면 그 안의 문제를 볼 수 있어요.
- product-ratio-cancellation 61% (11)
A long product whose factors are each a ratio of consecutive or near-consecutive values, such as (n+1)/n; writing the factors out shows every interior numerator cancelling a later denominator, so the product reduces to the surviving end numerators over the end denominators.
- difference-split-collapse 28% (5)
Each summand is first rewritten as f(n) minus f(n+1) using an identity — partial fractions, a logarithm of a ratio, or a factorial split — after which the sum collapses to the first f value minus the last, read off at the boundaries or as a limit.
- shifted-series-difference 11% (2)
Two sums whose terms are the same sequence offset by a fixed shift are subtracted; the overlapping middle block cancels in full and only the few unmatched terms at the head of one and the tail of the other remain.
더 보기 (연도 추세, 도구 fingerprint, 학년 분포, 전체 문제)
도구 fingerprint (1–17)
학년 분포
- Gr 4 1
- Gr 5 5
- Gr 6 1
- Gr 7 1