The entries in a certain row of Pascal's triangle are
1, n,...,n,1
The average of the entries in this row is 2048. Find n
i originally got 2047 but that was wrong
Let there be n + 1 entries in the row (including the 1s at the beginning and end). Since the average is 2048, the sum of all entries must be 2048 * (n + 1).
The entries in the given row are consecutive multiples of n, except for the 1s at the ends. Therefore, the sum can be expressed as 1 + \binom{n + 1}{1} + \binom{n + 1}{2} + ... + \binom{n + 1}{n + 1}.
This is an arithmetic series with n + 1 terms and a common difference of n. The sum of an arithmetic series is given by the formula: Sn = n/2 * (a1 + an), where Sn is the sum, n is the number of terms, a1 is the first term, and an is the last term.
In this case, a1 = 1 and an = \binom{n + 1}{n + 1}. Plugging these into the formula, we get:
Sn = (n + 1) / 2 * (1 + n^2)
Equating the sums: Now, equate the two expressions for the sum:
2048 * (n + 1) = (n + 1) / 2 * n^2
Cancel out the common factor (n + 1) on both sides:
2048 = (1 + n^2) / 2
Multiply both sides by 2:
4096 = 1 + n^2
Subtract 1 from both sides:
4095 = n^2
Take the square root of both sides:
63 = n
Therefore, n = 63 is the value that makes the average of the entries in the row equal to 2048.
The sum of the entries in any row = 2^n where the number of entries = n + 1 and n = 0,1,2,3,4.......
Therefore
2^n / (n + 1) = 2048
2^n = 2048 (n + 1) take the log of both sides
log (2^n) = log (2048 ( n+ 1) ) using base 2 log we can write
log2 (2^n) = log2 (2048 (n + 1) )
n log 2 2 = 11 + log2 (n + 1) { log 2 2 = 1 }
n = 11 + log2 (n + 1)
(n - 11) = log2 (n + 1) in exponential form we have
2^(n -11) = n + 1
Note that this will be true when n = 15