AP and GP Conditions Analysis

This problem requires us to find the sum of 5 consecutive terms in an Arithmetic Progression (AP). We are given the product of these 5 terms and a condition that the first, second, and fifth terms follow a Geometric Progression (GP).

Representing Consecutive AP Terms

To handle the conditions effectively, let's represent the 5 consecutive terms of the AP symmetrically around a central term \(a\), with a common difference \(d\). The terms can be written as:

• Term 1: \(a - 2d\)
• Term 2: \(a - d\)
• Term 3: \(a\)
• Term 4: \(a + d\)
• Term 5: \(a + 2d\)

The common difference of this AP is \(d\).

Applying the Product Condition

The product of these 5 terms is given as 229635.

\( (a - 2d)(a - d)(a)(a + d)(a + 2d) = 229635 \)

We can rearrange and use the difference of squares formula (\(x^2 - y^2 = (x-y)(x+y)\)) to simplify:

\( a \times [(a - d)(a + d)] \times [(a - 2d)(a + 2d)] = 229635 \)

\( a(a^2 - d^2)(a^2 - 4d^2) = 229635 \)

Let's refer to this as Equation (1).

Applying the GP Condition

The first term (\(a-2d\)), second term (\(a-d\)), and fifth term (\(a+2d\)) are in Geometric Progression (GP). This means the ratio between consecutive terms is constant:

\( \frac{a - d}{a - 2d} = \frac{a + 2d}{a - d} \)

To solve this, we cross-multiply:

\( (a - d)^2 = (a - 2d)(a + 2d) \)

Expand both sides:

\( a^2 - 2ad + d^2 = a^2 - 4d^2 \)

Simplify the equation by cancelling \(a^2\) from both sides and rearranging terms:

\( -2ad + d^2 = -4d^2 \)

\( 5d^2 = 2ad \)

This equation establishes a relationship between \(a\) and \(d\).

Solving for \(a\) and \(d\)

From the GP condition (\(5d^2 = 2ad\)), we analyze:

• Case 1: \(d = 0\) If the common difference \(d\) is 0, all terms are identical (\(a\)). The product equation becomes \(a^5 = 229635\). The fifth root of 229635 is approximately 13.15, which isn't a simple integer. This case usually leads to non-integer terms, so we proceed assuming \(d \ne 0\).
• Case 2: \(d \neq 0\) Since \(d \neq 0\), we can divide the equation \(5d^2 = 2ad\) by \(d\): \( 5d = 2a \) \( a = \frac{5}{2}d \) This provides a direct link between \(a\) and \(d\).

Now, we substitute \(a = \frac{5}{2}d\) into Equation (1) for the product:

\( a(a^2 - d^2)(a^2 - 4d^2) = 229635 \)

First, calculate the terms involving \(a^2\) in terms of \(d^2\):

• \(a^2 = (\frac{5}{2}d)^2 = \frac{25}{4}d^2\)
• \(a^2 - d^2 = \frac{25}{4}d^2 - d^2 = \frac{25d^2 - 4d^2}{4} = \frac{21}{4}d^2\)
• \(a^2 - 4d^2 = \frac{25}{4}d^2 - 4d^2 = \frac{25d^2 - 16d^2}{4} = \frac{9}{4}d^2\)

Substitute these expressions back into the product equation:

\( (\frac{5}{2}d) \times (\frac{21}{4}d^2) \times (\frac{9}{4}d^2) = 229635 \)

Multiply the constants and the powers of \(d\):

\( \frac{5 \times 21 \times 9}{2 \times 4 \times 4} d^{(1+2+2)} = 229635 \)

\( \frac{945}{32} d^5 = 229635 \)

Isolate \(d^5\):

\( d^5 = 229635 \times \frac{32}{945} \)

Perform the division and multiplication:

\( d^5 = 243 \times 32 \)

Recognize the numbers as powers:

\( d^5 = 3^5 \times 2^5 = (3 \times 2)^5 = 6^5 \)

This gives us the common difference:

\( d = 6 \)

Now, find the value of \(a\) using the relationship \(a = \frac{5}{2}d\):

\( a = \frac{5}{2} \times 6 = 5 \times 3 = 15 \)

Determining the Five Consecutive Terms

Using \(a = 15\) and \(d = 6\), we can list the 5 consecutive terms:

• Term 1: \(a - 2d = 15 - 2(6) = 15 - 12 = 3\)
• Term 2: \(a - d = 15 - 6 = 9\)
• Term 3: \(a = 15\)
• Term 4: \(a + d = 15 + 6 = 21\)
• Term 5: \(a + 2d = 15 + 2(6) = 15 + 12 = 27\)

The sequence of terms is 3, 9, 15, 21, 27.

We can quickly check if these terms satisfy the given conditions:

• They are in AP with a common difference of 6.
• Their product is \(3 \times 9 \times 15 \times 21 \times 27 = 229635\).
• The first (3), second (9), and fifth (27) terms are in GP, as \(9/3 = 3\) and \(27/9 = 3\).

The terms correctly match the problem statement.

Calculating the Sum of the Five Terms

The sum of the five consecutive terms is:

Sum = \((a - 2d) + (a - d) + a + (a + d) + (a + 2d)\)

Notice that the terms with \(d\) cancel each other out:

Sum = \(5a\)

Substitute the calculated value of \(a = 15\):

Sum = \(5 \times 15 = 75\)

Final Answer

The sum of all five terms is 75.