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d) Find [tex]$a_{25}$[/tex].
(5 marks)

e) Compute the summation given below when [tex]$n = 6$[/tex].
(5 marks)
[tex][tex]$
3 \sum_{k=1}^n(2k - 3) + \sum_{k=1}^n(4 - 5k)
$[/tex]

Sagot :

GinnyAnsweridn
Alright, let's tackle the summation problem step by step.

Given the expression:

[tex]\[ 3 \sum_{k=1}^n (2k - 3) + \sum_{k=1}^n (4 - 5k) \][/tex]

where [tex]\( n = 6 \)[/tex].

### Step 1: Calculate the First Summation Component

First, we consider the summation inside the first term:

[tex]\[ \sum_{k=1}^n (2k - 3) \][/tex]

Substitute [tex]\( n = 6 \)[/tex]:

[tex]\[ \sum_{k=1}^6 (2k - 3) \][/tex]

Calculate each term inside the summation for [tex]\( k = 1, 2, 3, 4, 5, 6 \)[/tex]:

- When [tex]\( k = 1 \)[/tex]: [tex]\( 2 \cdot 1 - 3 = -1 \)[/tex]
- When [tex]\( k = 2 \)[/tex]: [tex]\( 2 \cdot 2 - 3 = 1 \)[/tex]
- When [tex]\( k = 3 \)[/tex]: [tex]\( 2 \cdot 3 - 3 = 3 \)[/tex]
- When [tex]\( k = 4 \)[/tex]: [tex]\( 2 \cdot 4 - 3 = 5 \)[/tex]
- When [tex]\( k = 5 \)[/tex]: [tex]\( 2 \cdot 5 - 3 = 7 \)[/tex]
- When [tex]\( k = 6 \)[/tex]: [tex]\( 2 \cdot 6 - 3 = 9 \)[/tex]

Now add these values together:

[tex]\[ -1 + 1 + 3 + 5 + 7 + 9 = 24 \][/tex]

Thus, the first summation component is:

[tex]\[ \sum_{k=1}^6 (2k - 3) = 24 \][/tex]

Multiplying this by 3 gives:

[tex]\[ 3 \sum_{k=1}^6 (2k - 3) = 3 \cdot 24 = 72 \][/tex]

### Step 2: Calculate the Second Summation Component

Next, we consider the second summation:

[tex]\[ \sum_{k=1}^n (4 - 5k) \][/tex]

Substitute [tex]\( n = 6 \)[/tex]:

[tex]\[ \sum_{k=1}^6 (4 - 5k) \][/tex]

Calculate each term inside the summation for [tex]\( k = 1, 2, 3, 4, 5, 6 \)[/tex]:

- When [tex]\( k = 1 \)[/tex]: [tex]\( 4 - 5 \cdot 1 = -1 \)[/tex]
- When [tex]\( k = 2 \)[/tex]: [tex]\( 4 - 5 \cdot 2 = -6 \)[/tex]
- When [tex]\( k = 3 \)[/tex]: [tex]\( 4 - 5 \cdot 3 = -11 \)[/tex]
- When [tex]\( k = 4 \)[/tex]: [tex]\( 4 - 5 \cdot 4 = -16 \)[/tex]
- When [tex]\( k = 5 \)[/tex]: [tex]\( 4 - 5 \cdot 5 = -21 \)[/tex]
- When [tex]\( k = 6 \)[/tex]: [tex]\( 4 - 5 \cdot 6 = -26 \)[/tex]

Now add these values together:

[tex]\[ -1 + (-6) + (-11) + (-16) + (-21) + (-26) = -81 \][/tex]

Thus, the second summation component is:

[tex]\[ \sum_{k=1}^6 (4 - 5k) = -81 \][/tex]

### Step 3: Combine Both Summation Components

Finally, we combine the results of both summations:

[tex]\[ 3 \sum_{k=1}^6 (2k - 3) + \sum_{k=1}^6 (4 - 5k) = 72 + (-81) = -9 \][/tex]

### Final Answer

So, the value of the given summation when [tex]\( n = 6 \)[/tex] is:

[tex]\[ -9 \][/tex]
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