The 12 numbers, `a _(1), a _(2)………, a _(12)` are in arithmetical progression. The sum of al these numbers is 354. Let `P = a _(2) + a _(4) + ……………a _(12) and Q = a _(1) + a _(3) + ……..+a _(11) .` If the ratio P :Q is `32:27,` the common difference of the progression is

Answer

To solve the problem step by step, we will follow the given information about the arithmetic progression (AP) and the sums \( P \) and \( Q \).

### Step 1: Define the terms of the arithmetic progression
Let the first term of the arithmetic progression be \( a \) and the common difference be \( d \). The terms can be expressed as follows:
- \( a_1 = a \)
- \( a_2 = a + d \)
- \( a_3 = a + 2d \)
- ...
- \( a_{12} = a + 11d \)

### Step 2: Write the equation for the sum of the first 12 terms
The sum of the first 12 terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
For \( n = 12 \), we have:
\[ S_{12} = \frac{12}{2} \times (2a + 11d) = 6(2a + 11d) \]
Given that this sum is 354, we can set up the equation:
\[ 6(2a + 11d) = 354 \]
Dividing both sides by 6:
\[ 2a + 11d = 59 \quad \text{(Equation 1)} \]

### Step 3: Calculate \( P \) and \( Q \)
- \( P = a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} \)
\[ P = (a + d) + (a + 3d) + (a + 5d) + (a + 7d) + (a + 9d) + (a + 11d) = 6a + (1 + 3 + 5 + 7 + 9 + 11)d \]
The sum of the coefficients of \( d \) is \( 36 \):
\[ P = 6a + 36d \quad \text{(Equation 2)} \]

- \( Q = a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} \)
\[ Q = a + (a + 2d) + (a + 4d) + (a + 6d) + (a + 8d) + (a + 10d) = 6a + (0 + 2 + 4 + 6 + 8 + 10)d \]
The sum of the coefficients of \( d \) is \( 30 \):
\[ Q = 6a + 30d \quad \text{(Equation 3)} \]

### Step 4: Set up the ratio \( \frac{P}{Q} = \frac{32}{27} \)
Using Equations 2 and 3:
\[ \frac{6a + 36d}{6a + 30d} = \frac{32}{27} \]
Cross-multiplying gives:
\[ 27(6a + 36d) = 32(6a + 30d) \]
Expanding both sides:
\[ 162a + 972d = 192a + 960d \]
Rearranging gives:
\[ 162a - 192a = 960d - 972d \]
\[ -30a = -12d \]
Dividing both sides by -6:
\[ 5a = 2d \quad \text{(Equation 4)} \]

### Step 5: Substitute Equation 4 into Equation 1
Substituting \( a = \frac{2d}{5} \) into Equation 1:
\[ 2\left(\frac{2d}{5}\right) + 11d = 59 \]
This simplifies to:
\[ \frac{4d}{5} + 11d = 59 \]
Finding a common denominator:
\[ \frac{4d + 55d}{5} = 59 \]
\[ \frac{59d}{5} = 59 \]
Multiplying both sides by 5:
\[ 59d = 295 \]
Dividing by 59:
\[ d = 5 \]

### Final Answer
The common difference of the arithmetic progression is \( d = 5 \).

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