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Problem 661

# The 54 th and 4 th terms of an arithmetic progression are 61 and 64 ; find the $$23 \mathrm{rd}$$ term.

Expert verified
The 23rd term of the arithmetic progression is $$\frac{3143}{50}$$.
See the step by step solution

## Step 1: Understand the formula for arithmetic progressions

For an arithmetic progression, the formula to find the nth term is: $$a_n = a_1 + (n-1)d$$, where: - $$a_n$$ is the nth term of the progression - $$a_1$$ is the first term of the progression - n is the position of the term in the progression - d is the common difference between consecutive terms in the progression

## Step 2: Write the given information

We are given the 54th term and the 4th term. So, we have: $$a_{54} = 61$$ $$a_{4} = 64$$

## Step 3: Write the formulas for the given terms

Using the formula from Step 1, we can write the formulas for the given terms: $$a_{54} = a_1 + 53d = 61$$ $$a_{4} = a_1+3d = 64$$

## Step 4: Solve for the common difference d

Subtract the second equation from the first equation to eliminate $$a_1$$: $$(a_1 + 53d) - (a_1+3d) = 61 - 64$$ This simplifies to: $$50d = -3$$ Now, divide by 50: $$d = -\frac{3}{50}$$

## Step 5: Solve for the first term a1

Plug the value of d into one of the equations (here we use the equation for $$a_{4}$$): $$a_1 + 3\left(-\frac{3}{50}\right) = 64$$ Multiply out the term: $$a_1 - \frac{9}{50} = 64$$ Add the fraction to the other side: $$a_1 = 64 + \frac{9}{50}$$ Convert 64 to a fraction with the same denominator: $$a_1 = \frac{3200}{50} + \frac{9}{50}$$ Add the fractions together: $$a_1 = \frac{3209}{50}$$

## Step 6: Find the 23rd term a23

Now that we have the common difference and the first term, we can find the 23rd term using the formula from Step 1: $$a_{23} = a_1 + 22d$$ Plug in the values of $$a_1$$ and d: $$a_{23} = \frac{3209}{50} + 22\left(-\frac{3}{50}\right)$$ Multiply out the term: $$a_{23} = \frac{3209}{50} - \frac{66}{50}$$ Combine the fractions: $$a_{23} = \frac{3209-66}{50}$$ Simplify: $$a_{23} = \frac{3143}{50}$$ The 23rd term of the arithmetic progression is $$\frac{3143}{50}$$.

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