The average of 7 consecutive numbers is 20. The largest of these numbers is

### Concept & Logic For any sequence of consecutive numbers (an Arithmetic Progression), the average is always exactly equal to the median (the middle term). ### Step-by-Step Solution * **Given:** * There are $7$ consecutive numbers. * The average of these numbers is $20$. * **Deduction:** * Since there are $7$ terms (an odd number of terms), the middle term is the 4th term. * Therefore, the 4th term is exactly the average, which is $20$. * **Calculation:** * The sequence is centered around $20$. * The terms are: (1st), (2nd), (3rd), **$20$**, (5th), (6th), (7th) * Since they are consecutive integers, they increase by $1$. * 5th term = $21$ * 6th term = $22$ * 7th term (largest) = $23$ ### Exam Strategy & Shortcut If the average of $N$ consecutive integers is $A$ (where $N$ is odd), the largest number is simply $A + \frac{N - 1}{2}$. Largest = $20 + \frac{7 - 1}{2} = 20 + 3 = 23$. ### Common Pitfall Setting up a long algebraic equation: $x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 7 \times 20$. While this works ($7x + 21 = 140 \implies 7x = 119 \implies x = 17$, so largest is $17+6=23$), it is a massive waste of time for a problem that can be solved visually in 3 seconds. ### Final Answer **Therefore, the correct answer is 23.**