Let's break down the problem step by step:

Given Information:

• The sum of the first three numbers of an arithmetic progression (AP) is 60.
• The sum of the squares of these three numbers is 1250.
• We need to find the fifth term of the arithmetic progression (AP).

Step-by-Step Approach:

1. General Form of AP: Let the three terms of the AP be: $a - d, \, a, \, a + d$ where $a$ is the middle term, and $d$ is the common difference.

2. Sum of the First Three Terms: The sum of these three terms is given as: $(a - d) + a + (a + d) = 60$ Simplifying: $3a = 60 \quad \Rightarrow \quad a = 20$

3. Sum of Squares of the Three Terms: The sum of the squares of these terms is: $(a - d)^2 + a^2 + (a + d)^2 = 1250$ Expanding each term: $(20 - d)^2 + 20^2 + (20 + d)^2 = 1250$ $(400 - 40d + d^2) + 400 + (400 + 40d + d^2) = 1250$ Simplifying: $400 + 400 + 400 + 2d^2 = 1250$ $1200 + 2d^2 = 1250 \quad \Rightarrow \quad 2d^2 = 50 \quad \Rightarrow \quad d^2 = 25$ $d = 5$ (Taking the positive value of $d$ since it's an increasing AP.)

4. Finding the Fifth Term: The fifth term of the AP is given by: $a + 2d = 20 + 2(5) = 20 + 10 = 30$

Final Answer:

The fifth term of the arithmetic progression is 30.

Would you like further clarification on any step?

Related Questions:

1. How do we derive the general formula for the sum of the squares in an AP?
2. What happens if the common difference $d$ is negative?
3. Can you find the 10th term of the same arithmetic progression?
4. How do you solve this type of question if more terms are involved?
5. Can you explain the difference between geometric and arithmetic progressions?

Tip:

For arithmetic progressions, always remember that the sum of terms is dependent on both the first term and the common difference.