How many terms of the AP `20,19""1/3,18""2/3`,…., must be taken to make the sum 300 ? Explain the double answer.

Explanation

To find out how many terms of the arithmetic progression (AP) \(20, 19\frac{1}{3}, 18\frac{2}{3}, \ldots\) must be taken to make the sum equal to 300, we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \(a\) of the AP is:
\[ a = 20 \]

To find the common difference \(d\), we can subtract the first term from the second term:
\[ d = 19\frac{1}{3} - 20 = 19.33 - 20 = -0.67 = -\frac{2}{3} \]

### Step 2: Write the formula for the sum of the first \(n\) terms of an AP
The formula for the sum of the first \(n\) terms \(S_n\) of an AP is given by:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]

### Step 3: Set up the equation with the given sum
We know that \(S_n = 300\), so we can set up the equation:
\[ 300 = \frac{n}{2} \left(2 \times 20 + (n - 1) \left(-\frac{2}{3}\right)\right) \]

### Step 4: Simplify the equation
Substituting \(a\) and \(d\) into the equation:
\[ 300 = \frac{n}{2} \left(40 - \frac{2(n - 1)}{3}\right) \]
To eliminate the fraction, multiply both sides by 2:
\[ 600 = n \left(40 - \frac{2(n - 1)}{3}\right) \]
Now, multiply through by 3 to eliminate the denominator:
\[ 1800 = 3n \left(40 - \frac{2(n - 1)}{3}\right) \]
Distributing \(3n\):
\[ 1800 = 120n - 2n(n - 1) \]
This simplifies to:
\[ 1800 = 120n - 2n^2 + 2n \]
Rearranging gives:
\[ 2n^2 - 122n + 1800 = 0 \]

### Step 5: Solve the quadratic equation
Now we will solve the quadratic equation \(2n^2 - 122n + 1800 = 0\) using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 2\), \(b = -122\), and \(c = 1800\):
\[ b^2 - 4ac = (-122)^2 - 4 \times 2 \times 1800 = 14884 - 14400 = 484 \]
Now substituting back into the formula:
\[ n = \frac{122 \pm \sqrt{484}}{4} \]
Calculating \(\sqrt{484} = 22\):
\[ n = \frac{122 \pm 22}{4} \]
This gives us two possible solutions:
1. \(n = \frac{144}{4} = 36\)
2. \(n = \frac{100}{4} = 25\)

### Step 6: Conclusion
Thus, the number of terms that must be taken to make the sum 300 is either \(n = 25\) or \(n = 36\). The double answer arises because both the 25th and 36th terms yield the same sum of 300.