Question

In an arithmetic progression, if the sum of fourth, seventh and tenth terms is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is

Hint:

When given conditions on specific terms and on the sum of terms in an AP, convert them into equations using $T_n = a + (n-1)d$ and $S_n = \frac{n}{2}[2a + (n-1)d]$. Two independent conditions will usually give you two linear equations in $a$ and $d$.

Answer 1

Correct Answer: 65

Step 1: 4th+7th+10th terms \( =3a+18d=99 \Rightarrow a+6d=33 \).
Step 2: Sum of first 14 terms \( =\frac{14}{2}(2a+13d)=497 \Rightarrow 2a+13d=71 \).
Step 3: Solving the two equations: \( d=5 \), \( a=3 \).
Step 4: Sum of first 5 terms \( =\frac{5}{2}(2(3)+4(5))=\frac{5}{2}(26)=65 \).
\[ \boxed{65} \]