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.) If 7 times of the 7th term of an A.P. is equal to 11 times of its 11th term, then the 18th term of the A.P. is ____.

Hint:

For arithmetic progressions:
Always express conditions using \(a_n=a+(n-1)d\)
After finding \(a\) in terms of \(d\), substitute directly
Numerical answers should match the required decimal format

Answer 1

Correct Answer: 0

Step 1: Let the first term of the A.P. be \(a\) and the common difference be \(d\). The \(n^{\text{th}}\) term of an A.P. is: \[ a_n = a+(n-1)d \]
Step 2: Write the given condition. 7 times of the 7th term: \[ 7[a+6d] \] 11 times of the 11th term: \[ 11[a+10d] \] Given: \[ 7(a+6d)=11(a+10d) \]
Step 3: Simplify: \[ 7a+42d=11a+110d \] \[ -4a=68d \] \[ a=-17d \]
Step 4: Find the 18th term: \[ a_{18}=a+17d \] Substitute \(a=-17d\): \[ a_{18}=-17d+17d=0 \]
Step 5: Writing the answer up to two decimal places: \[ \boxed{0.00} \]