The question in the image is as follows:

"In an AP (Arithmetic Progression), the 16th term is three times the 5th term, and the 12th term is 100.
What is the sum of the first twenty terms?"

Let's solve this step-by-step.

Step 1: Formula for the nth term of an AP

The nth term of an AP is given by:

$T_n = a + (n - 1) d$

Where:

• $T_n$ is the nth term,
• $a$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

Step 2: Set up equations based on given conditions

1. Condition 1: 16th term is three times the 5th term.

   • 16th term: $T_{16} = a + 15d$
   • 5th term: $T_5 = a + 4d$

   According to the condition: $a + 15d = 3(a + 4d)$

2. Condition 2: The 12th term is 100.

   • 12th term: $T_{12} = a + 11d$
   • According to the condition: $a + 11d = 100$

Step 3: Solve the system of equations

We now have two equations:

1. $a + 15d = 3(a + 4d)$
2. $a + 11d = 100$

Let's solve these equations.

1. Simplify the first equation: $a + 15d = 3a + 12d$ $15d - 12d = 3a - a$ $3d = 2a$ $d = \frac{2a}{3}$

2. Substitute $d = \frac{2a}{3}$ into the second equation: $a + 11 \times \frac{2a}{3} = 100$ $a + \frac{22a}{3} = 100$ $\frac{3a + 22a}{3} = 100$ $\frac{25a}{3} = 100$ $25a = 300$ $a = 12$

3. Now substitute $a = 12$ back into $d = \frac{2a}{3}$: $d = \frac{2 \times 12}{3} = 8$

Step 4: Find the sum of the first 20 terms

The sum of the first $n$ terms of an AP is given by:

$S_n = \frac{n}{2} \times (2a + (n - 1)d)$

For $n = 20$, $a = 12$, and $d = 8$:

$S_{20} = \frac{20}{2} \times (2 \times 12 + (20 - 1) \times 8)$ $S_{20} = 10 \times (24 + 19 \times 8)$ $S_{20} = 10 \times (24 + 152)$ $S_{20} = 10 \times 176 = 1760$

Final Answer:

The sum of the first twenty terms is 1760.

Would you like any further clarifications or details? Here are a few related questions you might want to explore:

1. What is the general formula for the sum of an arithmetic progression?
2. How do the common difference and first term affect the progression of terms in an AP?
3. What is the significance of the 16th term in determining the behavior of the AP?
4. Can we apply this method to find the sum of different numbers of terms?
5. How does an arithmetic progression differ from a geometric progression?

Tip: Always remember that solving systems of equations is a powerful tool for determining unknowns in sequences!