The sum of the coefficients, the relations among them, and sums that use them. Then finding remainders, proving divisibility and approximating large powers.
3 lessons, about 1 h 15 min3 simulations11 problems3 practice questionsPython in every lesson
Created by StemPy LabThe coefficients in a row of Pascal's triangle obey a few simple rules. , Pascal's rule joins neighbours, and . The ratio of neighbours, , builds a whole row quickly.
Putting in gives for the sum of all the coefficients. Putting gives , so the odd-placed and even-placed terms each add to . The identity for turns the sum of into .
Write the number as a multiple of the divisor plus or minus , expand with the binomial theorem, and see which terms the divisor swallows. This gives remainders, divisibility and last digits. For a power with small , the first few terms are an excellent estimate.
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