The sum of all 2-digit numbers divisible by 5 is

### Concept & Formula A sequence of numbers with a constant difference forms an Arithmetic Progression (A.P.). To find the sum of an A.P., we use the formula: $$ S = \frac{n}{2}(a + l) $$ Where $n$ is the total number of terms, $a$ is the first term, and $l$ is the last term. ### Step-by-Step Solution * **Step 1:** Identify the A.P. series. The 2-digit numbers perfectly divisible by $5$ are: $10, 15, 20, \dots, 95$. Here, the first term $a = 10$, the last term $l = 95$, and the common difference $d = 5$. * **Step 2:** Calculate the total number of terms ($n$). Use the general $n$-th term formula: $l = a + (n-1)d$ $95 = 10 + (n-1)5$ $85 = 5(n-1)$ $17 = n - 1 \Rightarrow n = 18$. There are $18$ valid terms in total. * **Step 3:** Calculate the sum of the series. $S = \frac{18}{2} \times (10 + 95)$ $S = 9 \times 105$ $S = 945$. ### Exam Strategy & Shortcut **Average Rule:** The sum of any evenly spaced sequence is simply the average of the first and last terms multiplied by the total number of terms. Average $= \frac{10 + 95}{2} = 52.5$. The number of terms can be found mentally: $(95 - 10) \div 5 + 1 = 18$. Sum $= 52.5 \times 18 = 945$. This avoids writing out formal A.P. notation entirely. ### Common Pitfall The most common mistake is miscounting the number of terms in the set. Many students instinctively divide 100 by 5 to get 20 terms, forgetting that 5 (a 1-digit number) and 100 (a 3-digit number) must be explicitly excluded from the 2-digit constraint requested by the prompt. ### Final Answer **Therefore, the correct answer is 945.**