Evaluate ฮฃ๐r=1
16 (5r- 1)
We can start by factoring out the constant, 16:
ฮฃ๐r=1
16 (5r - 1) = 16 ฮฃ๐r=1 (5r - 1)
Next, we can use the formula for the sum of the first n natural numbers:
ฮฃ๐r=1 r = 1 + 2 + ... + n = n(n + 1)/2
Substituting this formula into our expression, we have:
16 ฮฃ๐r=1 (5r - 1) = 16 (5(1) - 1 + 5(2) - 1 + ... + 5n - 1)
= 16 [5(1 + 2 + ... + n) - n]
= 16 [5(n(n+1)/2) - n]
= 8n(5n-1)
Therefore, the expression ฮฃ๐r=1 16 (5r - 1) simplifies to 8n(5n-1).
To evaluate the sum ฮฃ๐๐=1 16 (5๐-1), we need to find the sum of the expression 16 (5๐-1) as ๐ takes on values from 1 to ๐.
Step 1: Substitute ๐ = 1 into the expression.
16 (5(1) - 1) = 16(5 - 1) = 16(4) = 64
Step 2: Substitute ๐ = 2 into the expression.
16 (5(2) - 1) = 16(10 - 1) = 16(9) = 144
Step 3: Continue this process until reaching ๐ = ๐.
For example, when ๐ = 3:
16 (5(3) - 1) = 16(15 - 1) = 16(14) = 224
Overall, the sum evaluates to the cumulative sum of each of these individual calculations:
64 + 144 + 224 + ... + (16(5๐ - 1))
If you provide a specific value for ๐, I can perform the necessary calculations to obtain the sum.