Arithmetic Progression And Quadratic Equations

Let the first term of the A.P. be and the common difference be . - The total number of terms is even, so let the total number of terms be . - The sum of odd terms is given as 24 and the sum of even terms is given as 30. - The last term exceeds the first term by , so we have the equation for the -th term: Simplifying, we get: Thus, Now, using the sum formula for an arithmetic progression: - The sum of the first odd terms is given by . - The sum of the first even terms is given by . By solving these equations, we can find that the total number of terms is .

Step 1: Understanding the Concept:
When arithmetic means are inserted between and , they form an A.P. with terms. The common difference is .
: Key Formula or Approach:
.
.
The -th mean is .
Step 2: Detailed Explanation:
The required mean is:
Mean
Mean
Mean .
Mean .
Step 3: Final Answer:
The mean of the specified arithmetic means is 134.

Concept: For a quadratic equation :

• Sum of roots
• Product of roots

For a geometric progression (G.P.): Step 1: {Find the sum and product of the roots.} Given equation: Here, Sum of roots: Product of roots: Step 2: {Find minimum value of the product.} Minimum occurs at : Step 3: {Find maximum value of the sum.} Maximum occurs when denominator is minimum: Step 4: {Form the geometric progression.} First term: Common ratio: Step 5: {Find the sum of first six terms.}

Step 1: Understanding the Question:
We are given the formula for the sum of the first terms ( ) of an Arithmetic Progression (A.P.).
We need to find the sum of the squares of the first 10 terms of this A.P., which is .
Step 2: Key Formula or Approach:
1. Find the -th term ( ) of the A.P. using the formula for , and .
2. Once we have the formula for , we need to calculate .
3. We will use the standard summation formulas:



Step 3: Detailed Explanation:
Given the sum of the first terms is .
First, let's find the -th term, .
For , .
For , the -th term is given by:
Let's check if this formula works for : , which matches . So the formula is valid for all .
Now, we need to find the sum of the squares of the first 10 terms:
Expand the term :
So the sum becomes:
Now we use the summation formulas with :
Substitute these values back into the expression for the sum:
Step 4: Final Answer:
The sum of squares of the first 10 terms of the A.P. is 15220.