Suppose `a_(1),a_(2)…` are in A.P. If `a_(8): a_(5)= 3:2,` then `a_(17) : a_(23)` is :

Options

A. `1:2`
B. `3:4`
C. `4:11`
D. `8:11`

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.).

### Step 1: Understand the terms in A.P.
In an arithmetic progression, the nth term can be expressed as:
\[ a_n = a + (n-1)d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 2: Set up the ratio given in the problem
We are given that:
\[ \frac{a_8}{a_5} = \frac{3}{2} \]

Using the formula for the nth term:
- \( a_8 = a + 7d \)
- \( a_5 = a + 4d \)

Thus, we can write:
\[ \frac{a + 7d}{a + 4d} = \frac{3}{2} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(a + 7d) = 3(a + 4d) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 2a + 14d = 3a + 12d \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate \( a \):
\[ 2a + 14d - 3a - 12d = 0 \]
This simplifies to:
\[ -a + 2d = 0 \]
Thus, we find:
\[ a = 2d \]

### Step 6: Find the ratio \( \frac{a_{17}}{a_{23}} \)
Now we need to find the ratio \( \frac{a_{17}}{a_{23}} \):
- \( a_{17} = a + 16d \)
- \( a_{23} = a + 22d \)

Substituting \( a = 2d \):
- \( a_{17} = 2d + 16d = 18d \)
- \( a_{23} = 2d + 22d = 24d \)

### Step 7: Set up the ratio
Now we can set up the ratio:
\[ \frac{a_{17}}{a_{23}} = \frac{18d}{24d} \]

### Step 8: Simplify the ratio
The \( d \) cancels out:
\[ \frac{18}{24} = \frac{3}{4} \]

### Final Answer
Thus, the final answer is:
\[ a_{17} : a_{23} = 3 : 4 \]

---