Question

For the following question, enter the correct numerical value up to TWO decimal places. (If the numerical value has more than two decimal places, round-off the value to TWO decimal places.) After inserting \(n\) A.M.’s between \(2\) and \(38\), the sum of the resulting progression is \(200\). The value of \(n\) is _____

Hint:

When arithmetic means are inserted:
Total number of terms \(=\) number of A.M.’s \(+2\)
Sum depends only on first term, last term, and number of terms
Always convert the final answer to the required decimal format

Answer 1

Correct Answer: 8

Step 1: If \(n\) arithmetic means are inserted between \(2\) and \(38\), then the total number of terms in the A.P. is: \[ n+2 \]
Step 2: First term: \[ a=2 \] Last term: \[ l=38 \]
Step 3: Sum of an arithmetic progression is: \[ S=\frac{n}{2}(a+l) \] Here, \[ S=200,\quad n \to (n+2) \] \[ 200=\frac{n+2}{2}(2+38) \] \[ 200=\frac{n+2}{2}\times 40 \] \[ 200=20(n+2) \]
Step 4: Solve for \(n\): \[ n+2=10 \] \[ n=8 \]
Step 5: Writing the answer up to two decimal places: \[ \boxed{8.00} \]