How many terms of the A.P. 17+15+13+… has the sum 72? Explain the double answer.

Explanation

To find how many terms of the arithmetic progression (A.P.) 17, 15, 13, ... have a sum of 72, we will follow these steps:

### Step 1: Identify the first term (A) and the common difference (d)
The first term \( A \) of the A.P. is 17. The common difference \( d \) can be calculated as:
\[ d = 15 - 17 = -2 \]

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first \( n \) terms \( S_n \) of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)d) \]
We know that \( S_n = 72 \), so we can set up the equation:
\[ 72 = \frac{n}{2} \times (2 \times 17 + (n - 1)(-2)) \]

### Step 3: Substitute the values into the equation
Substituting \( A = 17 \) and \( d = -2 \):
\[ 72 = \frac{n}{2} \times (34 - 2(n - 1)) \]
This simplifies to:
\[ 72 = \frac{n}{2} \times (34 - 2n + 2) \]
\[ 72 = \frac{n}{2} \times (36 - 2n) \]

### Step 4: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 144 = n(36 - 2n) \]
Expanding this gives:
\[ 144 = 36n - 2n^2 \]
Rearranging the equation leads to:
\[ 2n^2 - 36n + 144 = 0 \]
Dividing the entire equation by 2:
\[ n^2 - 18n + 72 = 0 \]

### Step 5: Solve the quadratic equation
To solve for \( n \), we can factor the quadratic:
\[ (n - 12)(n - 6) = 0 \]
Setting each factor to zero gives us:
\[ n - 12 = 0 \quad \Rightarrow \quad n = 12 \]
\[ n - 6 = 0 \quad \Rightarrow \quad n = 6 \]

### Step 6: Interpret the results
Thus, we have two possible values for \( n \): \( n = 6 \) and \( n = 12 \).

### Explanation of the double answer
The reason we have two values is that the sum of the first 6 terms equals 72, and the sum of the first 12 terms also equals 72. This can happen because the sum of the terms from the 7th to the 12th (which are negative) could sum to zero. Therefore, the sum of the first 6 terms is equal to the sum of the first 12 terms.

### Final Answer
The number of terms of the A.P. that sum to 72 is \( n = 6 \) and \( n = 12 \).

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