Sum of the First 212 Terms in Pascal's Triangle Sequence

Pascal's Triangle is a fascinating mathematical construct with a rich history and numerous applications in combinatorics and probability. Each row of Pascal's Triangle consists of binomial coefficients, which can be written out in a sequence. The problem at hand is to find the sum of the first 212 terms in this sequence. This requires a clear understanding of the structure of Pascal's Triangle and the application of series summation techniques.

The Structure of Pascal's Triangle

Pascal's Triangle is constructed such that the nth row contains n 1 terms. The kth term in the nth row is represented by the binomial coefficient Binom(n, k), where k 0, 1, 2, ..., n. The total number of terms in the first n rows can be calculated using the formula for the sum of the first n integers:

Total terms up to row n (n * (n 1)) / 2

Our goal is to determine the largest n such that the total number of terms up to row n is less than or equal to 212.

Determining the Number of Rows

Mathematically, we solve the inequality:

(n * (n 1)) / 2 ≤ 212

Multiplying both sides by 2, we get:

n * (n 1) ≤ 424

Testing integer values for n:

For n 20: (20 * 21) 420, valid but not large enough. For n 19: (19 * 20) 380, valid.

The total number of terms in the first 18 rows is 190. Our goal is to sum the terms from row 0 to row 18, and then add the terms from the remaining rows up to the 212th term.

The Sum of Binomial Coefficients in Each Row

The sum of the terms in row n is given by the formula:

2^n

Thus, the sum of all terms from row 0 to row 18 can be calculated using the sum of a geometric series:

S 1 * ((2^19 - 1) / (2 - 1)) 2^19 - 1

Calculating 2^19 we get:

2^19 524,288

Therefore, the sum of the terms from row 0 to row 18 is:

2^19 - 1 524,287

Calculating the Remaining Terms

We need to add the sum of the first 22 terms of row 19 to the total calculated above. The first 22 terms of row 19 are given as:

1, 19, 171, 969, 3876, 11628, 27132, 50388, 75582, 92378, 92378, 75582, 50388, 27132, 11628, 3876, 969, 171, 19, 1

The sum of the first 10 terms and the next 12 terms (symmetric terms in reverse) are:

1 19 171 969 3876 11628 27132 50388 75582 92378 167,960 92378 75582 50388 27132 11628 3876 969 171 19 1 167,960

The total sum for the first 22 terms is:

167,960 167,960 335,920

Therefore, the total sum of the first 212 terms is:

524,287 335,920 860,207

Conclusion

The sum of the first 212 terms in the Pascal's Triangle sequence is:

860,207