Arithmetic Progression

Step 1: Let the A.P. have first term and common difference . Sum of first terms:
Step 2: Given: Substitute the formula for sums: Cancel :
Step 3: Cross-multiply: Simplifying: Since ,
Step 4: General term of the A.P.:
Step 5: Compute the required ratio:

Step 1: Given first term . Let common difference be . Sum of first terms of an A.P.:
Step 2: Sum of first five terms:
Step 3: Sum of first ten terms: Sum of next five terms:
Step 4: Given condition:
Step 5: Now find the sum of first terms:

Step 1: If arithmetic means are inserted between and , then the total number of terms in the A.P. is:
Step 2: First term: Last term:
Step 3: Sum of an arithmetic progression is: Here,
Step 4: Solve for :
Step 5: Writing the answer up to two decimal places:

Step 1: Let the first term of the A.P. be and the common difference be . The term of an A.P. is:
Step 2: Write the given condition. 7 times of the 7th term: 11 times of the 11th term: Given:
Step 3: Simplify:
Step 4: Find the 18th term: Substitute :
Step 5: Writing the answer up to two decimal places: