Find the sum of first seven numbers which are multiples of 2 as well as of 9.
504
Step 1: We need numbers that are multiples of both 2 and 9.
The smallest number that is a multiple of both 2 and 9 is their Least Common Multiple (LCM).
LCM of 2 and 9 is:
2 = \(2\)
9 = \(3 \times 3\)
LCM = \(2 \times 3 \times 3 = 18\).
Step 2: So, the first number which is a multiple of both 2 and 9 is 18.
The sequence of such numbers will be: \(18, 36, 54, 72, 90, 108, 126, \ldots\)
Step 3: We only need the first 7 terms:
\(18, 36, 54, 72, 90, 108, 126\).
Step 4: This is an Arithmetic Progression (AP) because the difference between terms is constant.
Common difference (d) = \(36 - 18 = 18\).
Step 5: Formula for the sum of first \(n\) terms of an AP is:
\[ S_n = \dfrac{n}{2} \times (a + l) \]
where,
\(n = 7\) (number of terms),
\(a = 18\) (first term),
\(l = 126\) (last term).
Step 6: Substitute the values:
\[ S_7 = \dfrac{7}{2} \times (18 + 126) \]
\[ S_7 = \dfrac{7}{2} \times 144 \]
\[ S_7 = 7 \times 72 \]
\[ S_7 = 504 \]
Final Answer: The sum of the first seven numbers which are multiples of both 2 and 9 is 504.