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The 16th term of an arithmetic progression (AP) is 1 more than twice its 8th term. If the 12th term of the AP is 47, find its nth term.

Answer :

Final answer:

The 16th term of an Arithmetic Progression (AP) is 1 more than twice its 8th term. Given that the 12th term is 47, we need to find the nth term of the AP. To do this, we first find the common difference of the AP and then use the formula for the nth term of an AP to find the value.

Explanation:

In this question, we are given that the 16th term of an Arithmetic Progression (AP) is 1 more than twice its 8th term. We are also given that the 12th term of the AP is 47. We need to find the nth term of the AP.

Let's first find the common difference of the AP. To do that, we can subtract the 8th term from the 16th term and divide by the difference in their positions:

(16th term - 8th term) / (16 - 8) = (2nd term - 1st term) / (2 - 1)

Using the given information, we can substitute the values into the equation:

(16th term - 8th term) / 8 = (2nd term - 1st term) / 1

Simplifying the equation, we get:

(16th term - 8th term) / 8 = 2nd term - 1st term

Substituting the known values, we have:

(16th term - 47) / 8 = (2nd term - 1st term)

Let's substitute the known value of the 12th term into the equation to solve for the common difference:

(12th term - 47) / 4 = (2nd term - 1st term)

Simplifying further, we get:

(2nd term - 1st term) = (12th term - 47) / 4

Now, we can substitute the value of (2nd term - 1st term) into the equation we derived earlier:

(16th term - 47) / 8 = (12th term - 47) / 4

We can now solve for the 16th term by multiplying both sides of the equation by 8:

16th term - 47 = 2 * (12th term - 47)

Expanding the equation, we get:

16th term - 47 = 2 * 12th term - 2 * 47

Simplifying further, we have:

16th term - 47 = 2 * 12th term - 94

Adding 47 to both sides of the equation, we get:

16th term = 2 * 12th term - 47 + 47

Simplifying, we have:

16th term = 2 * 12th term

Substituting the known value of the 12th term, we get:

16th term = 2 * 47

Simplifying, we have:

16th term = 94

Therefore, the 16th term of the AP is 94. Now, let's find the common difference. We can subtract the 2nd term from the 1st term:

2nd term - 1st term = 94 - 47

Simplifying, we have:

2nd term - 1st term = 47

Therefore, the common difference of the AP is 47. Since we know the 1st term of the AP is 47, we can now write a general formula for the nth term of the AP:

nth term = 47 + (n - 1) * 47

This formula will give us the value of the nth term in the AP.

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