An expression contains 26 terms, one for each letter of the alphabet. It starts
a + 4b + 9c + 16d + 25e + … Another expression containing 26 terms starts a + 2b + 4c + 8d + 16e + … What is the sum of all
the coefficients?
the coefficients are just powers of 2, a geometric sequence.
Sn = a(2^n -1)/(2-1)
To find the sum of all the coefficients, we need to calculate the value of each term and then add them together. Let's calculate the value of each term for both expressions:
For the first expression: a + 4b + 9c + 16d + 25e + ...
To find the value of each term, we can use the formula for the sum of squares: n(n+1)(2n+1)/6, where n represents the position of the term in the alphabet (starting from 1).
For example:
- The term "a" is in the first position, so its value is 1(1+1)(2(1)+1)/6 = 1.
- The term "b" is in the second position, so its value is 2(2+1)(2(2)+1)/6 = 8.
- The term "c" is in the third position, so its value is 3(3+1)(2(3)+1)/6 = 27/2 = 13.5.
- The term "d" is in the fourth position, so its value is 4(4+1)(2(4)+1)/6 = 20.
- The term "e" is in the fifth position, so its value is 5(5+1)(2(5)+1)/6 = 55/2 = 27.5.
Similarly, we can calculate the value of all the other terms in the expression.
For the second expression: a + 2b + 4c + 8d + 16e + ...
In this case, each term is simply twice the value of the corresponding term in the first expression. So, we can multiply each value from the first expression by 2 to get the value of each term in the second expression.
Once we have calculated the value of each term in both expressions, we can add them together to find the sum of all the coefficients.