The 5th term of an arithmetic progression is 3times of 2nd term and 12th term exceeds 2times of 6th term by 1. Find the 16th term

What do you think the answer is?

To solve this problem, we need to use the given information to form equations and then solve them. Let's break it down step by step.

Step 1: Identify the given information:
- The 5th term of the arithmetic progression is 3 times the 2nd term.
- The 12th term exceeds 2 times the 6th term by 1.

Step 2: Use the information to form equations:
Let's assume that the first term of the arithmetic progression is "a", and the common difference is "d".

Using the first piece of information, we can form an equation:
5th term = 2nd term * 3
(a + 4d) = 3(a + d)

Using the second piece of information, we can form another equation:
12th term = 2 * 6th term + 1
(a + 11d) = 2(a + 5d) + 1

Step 3: Solve the equations:
Let's solve the equations to find the values of "a" and "d".

Equation 1: (a + 4d) = 3(a + d)
Expand the equation: a + 4d = 3a + 3d
Move terms to one side: 2a = d

Equation 2: (a + 11d) = 2(a + 5d) + 1
Expand the equation: a + 11d = 2a + 10d + 1
Move terms to one side: a - d = 1

Since we have two equations with two variables, we can solve them simultaneously.

Equation 1: 2a = d
Equation 2: a - d = 1

From Equation 2, we can solve for "a":
a = d + 1

Substituting the value of "a" in Equation 1:
2(d + 1) = d
2d + 2 = d
2 = d

Now that we know the value of "d", which is the common difference, we can find the value of "a" by substituting it into Equation 2:
a = d + 1
a = 2 + 1
a = 3

So, the first term of the arithmetic progression is 3, and the common difference is 2.

Step 4: Find the 16th term:
We now have the values of the first term (a = 3) and the common difference (d = 2). We can use the formula for the nth term of an arithmetic progression to find the 16th term.

The formula for the nth term of an arithmetic progression is given by:
nth term = a + (n - 1) * d

Substituting the values:
16th term = 3 + (16 - 1) * 2
16th term = 3 + 15 * 2
16th term = 3 + 30
16th term = 33

Therefore, the 16th term of the arithmetic progression is 33.