Statistics

Step 1: Rewrite the given differential equation in standard linear form:
Step 2: Identify the integrating factor (I.F.):
Step 3: Multiply the entire equation by the integrating factor:
Step 4: Observe that the left-hand side is the derivative of :
Step 5: Integrate both sides:
Step 6: Solve for :
Step 7: Use the condition :
Step 8: Substitute :
Step 9: Evaluate :

Step 1: The sequence contains terms in A.P. with first term and last term .
Step 2: Mean of an A.P.:
Step 3: Deviations from mean are symmetric about the central term. For an A.P. of odd number of equally spaced terms , mean deviation = average of .
Step 4:
Step 5: Equate to 255: Hence → (D).

Step 1: Recall the relation between mean and total sum. If the mean of items is , then the sum of the items is: Step 2: Find the total increment added to all items. Each item is increased by: This is a geometric progression with: Sum of this G.P.: Step 3: Find the increase in mean. Increase in mean: Step 4: Write the new mean. Step 5: Final conclusion. The correct option is .