Let a1, a2, a3 ldots ldots an be an increasing A.P. of positive integers such that a3=10, then if the maximum value of S= displaystyle∑n=14 aan is M. Find the sum of digits of ((M/10)).

Question

Let a1, a2, a3 ldots ldots an be an increasing A.P. of positive integers such that a3=10, then if the maximum value of S= displaystyle∑n=14 aan is M. Find the sum of digits of ((M/10)). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

a1=10-2 d ≥ 1 ⇒ d ≤ (9/2) d>0 S=aa1+aa2+aa3+aa4 =a1+(a1-1) d+a1+(a2-1) d+a1+(a3-1) d+a1+(a4-1) d =4 a1+d(a1+a2+a3+a4-4) =4 a1+d(4 a1+6 d-4) =4 a1(1+d)+6 d2-4 d=4(10-2 d)(1+d)+6 d2-4 d =40-28 d-2 d2 =40-2(d2-14 d+48)+98 =138-2(d-7)2 text for maximum sum d=4 for maximum sum d=4 M = 120 ∴[( M /10)]=12 ⇒ sum of digits =3

Q. Let a1​,a2​,a3​……an​ be an increasing A.P. of positive integers such that a3​=10, then if the maximum value of S=n=1∑4​aan​​ is M. Find the sum of digits of (10M​).

Q. Let $a_{1}, a_{2}, a_{3} \dots\dots a_{n}$ be an increasing A.P. of positive integers such that $a_{3} = 10$ , then if the maximum value of $S = n = 1 \sum 4 a_{a_{n}}$ is $M$ . Find the sum of digits of $(\frac{M}{10})$ .