Question $\sum_{k=1}^{n}\left(b_{k}-9 a_{k}\right)$ where $a_k = 7$ and $b_k = 30$ .

Studdy Solution

STEP 1

1. The summation is over the index $k$ from $1$ to $n$ .
2. The terms $a_k$ and $b_k$ are constants for each $k$ , given by $a_k = 7$ and $b_k = 30$ .
3. We need to evaluate the summation expression given these constants.

STEP 2

1. Substitute the constants $a_k$ and $b_k$ into the expression.
2. Simplify the expression inside the summation.
3. Evaluate the summation for the simplified expression.

STEP 3

Substitute the given constants $a_k = 7$ and $b_k = 30$ into the expression $b_k - 9a_k$ :
$b_k - 9a_k = 30 - 9 \times 7$

STEP 4

Simplify the expression $30 - 9 \times 7$ :
$30 - 63 = -33$
Thus, each term in the summation is $-33$ .

STEP 5

Evaluate the summation $\sum_{k=1}^{n} (-33)$ :
Since $-33$ is a constant, the summation simplifies to:
$\sum_{k=1}^{n} (-33) = -33 \times n$
The final result of the summation is $-33n$ .

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