to find nth term of the question 2/5, 3/10, 4/17, 5/26

To find the nth term of the sequence you've provided—2/5, 3/10, 4/17, 5/26—we need to analyze the pattern in both the numerators and the denominators separately. Let's break it down step by step.

Identifying Patterns

First, let’s look at the numerators: 2, 3, 4, 5. It’s clear that these numbers are increasing by 1 each time. So, we can express the numerator in terms of n as:

• Numerator = n + 1

Now, let’s examine the denominators: 5, 10, 17, 26. This sequence is not as straightforward, so we need to find a pattern here as well. Here are the differences between the consecutive terms:

• 10 - 5 = 5
• 17 - 10 = 7
• 26 - 17 = 9

The differences (5, 7, 9) are increasing by 2 each time, which suggests that the denominators are following a quadratic pattern. We can use the formula for the nth term of a quadratic sequence:

Finding the Denominator Formula

Let’s denote the denominator as D(n). Since the second difference is constant (2), we can infer that the general form of the denominator is:

• D(n) = an^2 + bn + c

To find the coefficients a, b, and c, we can set up a system of equations based on known values of D(n):

• D(1) = 5: a(1)^2 + b(1) + c = 5
• D(2) = 10: a(2)^2 + b(2) + c = 10
• D(3) = 17: a(3)^2 + b(3) + c = 17

This gives us three equations:

• a + b + c = 5 (1)
• 4a + 2b + c = 10 (2)
• 9a + 3b + c = 17 (3)

Now, we can solve this system:

Step 1: Eliminate c

Subtract equation (1) from (2) and (3):

• From (2) - (1): 3a + b = 5 (4)
• From (3) - (1): 8a + 2b = 12 (5)

Step 2: Simplify

Now, simplify equation (5) by dividing by 2:

• 4a + b = 6 (6)

Step 3: Solve for a and b

Now we have two equations:

• 3a + b = 5 (4)
• 4a + b = 6 (6)

Subtract equation (4) from (6):

• a = 1

Now substitute a = 1 back into equation (4):

• 3(1) + b = 5 → b = 2

Now plug a and b back into equation (1) to find c:

• 1 + 2 + c = 5 → c = 2

Final Form for the Denominator

Now that we have a, b, and c, we can express the denominator:

• D(n) = n^2 + 2n + 2

Putting it All Together

Now that we have both the numerator and the denominator, we can write the nth term of the sequence:

• Term(n) = (n + 1) / (n^2 + 2n + 2)

So, to find the nth term in your sequence, simply plug in the value of n into this formula. For example:

• n = 1: (1 + 1) / (1^2 + 2(1) + 2) = 2 / 5
• n = 2: (2 + 1) / (2^2 + 2(2) + 2) = 3 / 10

This method will give you the correct term for any value of n in the sequence. If you have any further questions or need clarification on any part, feel free to ask!

dear student,