Expression containing 26 terms starts at a + 2b +4c +8d +16 e. What is the sum of all coefficients?
forget the variables. You just want the coefficients, which form the GP
1, 2, 4, ... 2^25
Sn = a(r^n - 1)/(r-1)
32 f
32 f 64 g 128h 256 i 526j k1024 l 2048
yes
no
i dont know
To find the sum of all coefficients in the given expression, you need to first understand the pattern of how the coefficients are changing.
The expression starts with the term \(a + 2b + 4c + 8d + 16e\). The first coefficient is 1 (for \(a\)). Each subsequent coefficient is obtained by doubling the previous one. So, the second coefficient for term \(b\) is \(2\), the third coefficient for term \(c\) is \(4 \times 2 = 8\), the fourth coefficient for term \(d\) is \(8 \times 2 = 16\), and so on.
In this pattern, the coefficient for term \(e\) is \(16 \times 2 = 32\). The 26th coefficient can be found by continuing the pattern: \(32 \times 2 = 64\).
Now, let's find the sum of all coefficients. We can utilize the formula for the sum of a geometric series:
\[S = \frac{a(1 - r^n)}{1 - r}\]
Here, \(a\) is the first term in the series, \(r\) is the common ratio, and \(n\) is the number of terms.
In our case, the first term is 1, the common ratio is 2, and the number of terms is 26. Substituting these values into the formula, we have:
\[S = \frac{1(1 - 2^{26})}{1 - 2}\]
\[S = \frac{1 - 2^{26}}{-1}\]
\[S = 2^{26} - 1\]
Therefore, the sum of all coefficients in the expression is \(2^{26} - 1\), which is equal to 67,108,863.