Full solution

Q. To find the largest natural number less than

100

that is equal to the sum of

6

consecutive natural numbers, we can start by looking at the sum of the first few consecutive natural numbers.

Denote Smallest Number: Let's denote the smallest of the $6$ consecutive natural numbers as $n$ . Then the consecutive numbers can be represented as $n, n+1, n+2, n+3, n+4$ , and $n+5$ . The sum of these numbers is the arithmetic series sum formula: $S = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5)$ .
Simplify Sum Formula: To simplify the sum, we combine like terms: $S = 6n + (1+2+3+4+5) = 6n + 15$ .
Set Up Inequality: We want to find the largest value of $S$ that is less than $100$ . To do this, we can set up an inequality: $6n + 15 < 100$ .
Solve for n: Subtract $15$ from both sides of the inequality to solve for $n$ : $6n < 85$ .
Find Largest n Value: Divide both sides of the inequality by $6$ to find the largest possible value for $n$ : $n < \frac{85}{6}$ .
Substitute $n$ into Formula: Calculating $\frac{85}{6}$ gives us approximately $14.16$ . Since $n$ must be a natural number, we take the largest whole number less than $14.16$ , which is $14$ .
Calculate Largest Sum: Now we substitute $n = 14$ back into the sum formula to find the largest sum: $S = 6(14) + 15 = 84 + 15 = 99$ .
Check Sum Validity: We check to ensure that $99$ is indeed the sum of $6$ consecutive natural numbers: $14 + 15 + 16 + 17 + 18 + 19 = 99$ .