Answer :
To determine which of the given summations represent the series 11, 17, 23, and 29, we need to evaluate each summation for the specified values of [tex]\( n \)[/tex].
Let's consider each summation one by one:
### 1. [tex]\(\sum_{n=1}^4(5n+6)\)[/tex]
Evaluate the expression [tex]\(5n + 6\)[/tex] for [tex]\(n = 1, 2, 3, \text{and} 4\)[/tex]:
- For [tex]\(n = 1: \qquad 5(1) + 6 = 5 + 6 = 11\)[/tex]
- For [tex]\(n = 2: \qquad 5(2) + 6 = 10 + 6 = 16\)[/tex]
- For [tex]\(n = 3: \qquad 5(3) + 6 = 15 + 6 = 21\)[/tex]
- For [tex]\(n = 4: \qquad 5(4) + 6 = 20 + 6 = 26\)[/tex]
The resulting series is [tex]\(11, 16, 21, 26\)[/tex], which does not match [tex]\(11, 17, 23, 29\)[/tex]. Therefore, this summation does not represent the series.
### 2. [tex]\(\sum_{n=1}^4(6n+5)\)[/tex]
Evaluate the expression [tex]\(6n + 5\)[/tex] for [tex]\(n = 1, 2, 3, \text{and} 4\)[/tex]:
- For [tex]\(n = 1: \qquad 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 2: \qquad 6(2) + 5 = 12 + 5 = 17\)[/tex]
- For [tex]\(n = 3: \qquad 6(3) + 5 = 18 + 5 = 23\)[/tex]
- For [tex]\(n = 4: \qquad 6(4) + 5 = 24 + 5 = 29\)[/tex]
The resulting series is [tex]\(11, 17, 23, 29\)[/tex], which matches the target series. Thus, this summation represents the series.
### 3. [tex]\(\sum_{n=0}^3(6(n+1)+5)\)[/tex]
Evaluate the expression [tex]\(6(n+1) + 5\)[/tex] for [tex]\(n = 0, 1, 2, \text{and} 3\)[/tex]:
- For [tex]\(n = 0: \qquad 6(0+1) + 5 = 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 1: \qquad 6(1+1) + 5 = 6(2) + 5 = 12 + 5 = 17\)[/tex]
- For [tex]\(n = 2: \qquad 6(2+1) + 5 = 6(3) + 5 = 18 + 5 = 23\)[/tex]
- For [tex]\(n = 3: \qquad 6(3+1) + 5 = 6(4) + 5 = 24 + 5 = 29\)[/tex]
The resulting series is [tex]\(11, 17, 23, 29\)[/tex], which matches the target series. Thus, this summation represents the series.
### 4. [tex]\(\sum_{n=0}^3(6(n-1)+5)\)[/tex]
Evaluate the expression [tex]\(6(n-1) + 5\)[/tex] for [tex]\(n = 0, 1, 2, \text{and} 3\)[/tex]:
- For [tex]\(n = 0: \qquad 6(0-1) + 5 = 6(-1) + 5 = -6 + 5 = -1\)[/tex]
- For [tex]\(n = 1: \qquad 6(1-1) + 5 = 6(0) + 5 = 0 + 5 = 5\)[/tex]
- For [tex]\(n = 2: \qquad 6(2-1) + 5 = 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 3: \qquad 6(3-1) + 5 = 6(2) + 5 = 12 + 5 = 17\)[/tex]
The resulting series is [tex]\(-1, 5, 11, 17\)[/tex], which does not match [tex]\(11, 17, 23, 29\)[/tex]. Therefore, this summation does not represent the series.
### Conclusion
The summations that represent the series [tex]\(11, 17, 23, 29\)[/tex] are:
- [tex]\(\sum_{n=1}^4(6n+5)\)[/tex]
- [tex]\(\sum_{n=0}^3(6(n+1)+5)\)[/tex]
Let's consider each summation one by one:
### 1. [tex]\(\sum_{n=1}^4(5n+6)\)[/tex]
Evaluate the expression [tex]\(5n + 6\)[/tex] for [tex]\(n = 1, 2, 3, \text{and} 4\)[/tex]:
- For [tex]\(n = 1: \qquad 5(1) + 6 = 5 + 6 = 11\)[/tex]
- For [tex]\(n = 2: \qquad 5(2) + 6 = 10 + 6 = 16\)[/tex]
- For [tex]\(n = 3: \qquad 5(3) + 6 = 15 + 6 = 21\)[/tex]
- For [tex]\(n = 4: \qquad 5(4) + 6 = 20 + 6 = 26\)[/tex]
The resulting series is [tex]\(11, 16, 21, 26\)[/tex], which does not match [tex]\(11, 17, 23, 29\)[/tex]. Therefore, this summation does not represent the series.
### 2. [tex]\(\sum_{n=1}^4(6n+5)\)[/tex]
Evaluate the expression [tex]\(6n + 5\)[/tex] for [tex]\(n = 1, 2, 3, \text{and} 4\)[/tex]:
- For [tex]\(n = 1: \qquad 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 2: \qquad 6(2) + 5 = 12 + 5 = 17\)[/tex]
- For [tex]\(n = 3: \qquad 6(3) + 5 = 18 + 5 = 23\)[/tex]
- For [tex]\(n = 4: \qquad 6(4) + 5 = 24 + 5 = 29\)[/tex]
The resulting series is [tex]\(11, 17, 23, 29\)[/tex], which matches the target series. Thus, this summation represents the series.
### 3. [tex]\(\sum_{n=0}^3(6(n+1)+5)\)[/tex]
Evaluate the expression [tex]\(6(n+1) + 5\)[/tex] for [tex]\(n = 0, 1, 2, \text{and} 3\)[/tex]:
- For [tex]\(n = 0: \qquad 6(0+1) + 5 = 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 1: \qquad 6(1+1) + 5 = 6(2) + 5 = 12 + 5 = 17\)[/tex]
- For [tex]\(n = 2: \qquad 6(2+1) + 5 = 6(3) + 5 = 18 + 5 = 23\)[/tex]
- For [tex]\(n = 3: \qquad 6(3+1) + 5 = 6(4) + 5 = 24 + 5 = 29\)[/tex]
The resulting series is [tex]\(11, 17, 23, 29\)[/tex], which matches the target series. Thus, this summation represents the series.
### 4. [tex]\(\sum_{n=0}^3(6(n-1)+5)\)[/tex]
Evaluate the expression [tex]\(6(n-1) + 5\)[/tex] for [tex]\(n = 0, 1, 2, \text{and} 3\)[/tex]:
- For [tex]\(n = 0: \qquad 6(0-1) + 5 = 6(-1) + 5 = -6 + 5 = -1\)[/tex]
- For [tex]\(n = 1: \qquad 6(1-1) + 5 = 6(0) + 5 = 0 + 5 = 5\)[/tex]
- For [tex]\(n = 2: \qquad 6(2-1) + 5 = 6(1) + 5 = 6 + 5 = 11\)[/tex]
- For [tex]\(n = 3: \qquad 6(3-1) + 5 = 6(2) + 5 = 12 + 5 = 17\)[/tex]
The resulting series is [tex]\(-1, 5, 11, 17\)[/tex], which does not match [tex]\(11, 17, 23, 29\)[/tex]. Therefore, this summation does not represent the series.
### Conclusion
The summations that represent the series [tex]\(11, 17, 23, 29\)[/tex] are:
- [tex]\(\sum_{n=1}^4(6n+5)\)[/tex]
- [tex]\(\sum_{n=0}^3(6(n+1)+5)\)[/tex]
