If `a_(1)=2` and `a_(n)=2a_(n-1)+5` for `ngt1`, the value of `sum_(r=2)^(5)a_(r)` is

Options

A. 130
B. 160
C. 190
D. 220

Answer

To solve the problem step by step, we need to find the values of \( a_2, a_3, a_4, \) and \( a_5 \) using the recurrence relation provided, and then sum these values.

### Step 1: Find \( a_2 \)
Given:
- \( a_1 = 2 \)
- The recurrence relation: \( a_n = 2a_{n-1} + 5 \)

For \( n = 2 \):
\[ a_2 = 2a_1 + 5 = 2 \cdot 2 + 5 = 4 + 5 = 9 \]

### Step 2: Find \( a_3 \)
For \( n = 3 \):
\[ a_3 = 2a_2 + 5 = 2 \cdot 9 + 5 = 18 + 5 = 23 \]

### Step 3: Find \( a_4 \)
For \( n = 4 \):
\[ a_4 = 2a_3 + 5 = 2 \cdot 23 + 5 = 46 + 5 = 51 \]

### Step 4: Find \( a_5 \)
For \( n = 5 \):
\[ a_5 = 2a_4 + 5 = 2 \cdot 51 + 5 = 102 + 5 = 107 \]

### Step 5: Calculate the sum \( \sum_{r=2}^{5} a_r \)
Now, we need to find the sum:
\[ \sum_{r=2}^{5} a_r = a_2 + a_3 + a_4 + a_5 = 9 + 23 + 51 + 107 \]

Calculating the sum:
\[ 9 + 23 = 32 \]
\[ 32 + 51 = 83 \]
\[ 83 + 107 = 190 \]

### Final Answer
The value of \( \sum_{r=2}^{5} a_r \) is \( 190 \).

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