Latest Bank Exam Aptitude Question SOLUTION: displaystyle{text{Number of pairs * Sum of each pair} = (frac{n}{2})(n+1) = frac{n(n+1)}{2}} which is the formula above. Wait — what about an odd number of items? Ah, I’m glad

• Placement Papers
• /
• Bank Exam
• /
• Numerical Ability

Bank Exam Question Papers:

(#M40164994) BANK EXAM Numerical Ability question TRICK Keep an EYE

displaystyle{text{Number of pairs * Sum of each pair} = (frac{n}{2})(n+1) = frac{n(n+1)}{2}}
which is the formula above.

Wait — what about an odd number of items?
Ah, I’m glad you brought it up. What if we are adding up the numbers 1 to 9? We don’t have an even number of items to pair up. Many explanations will just give the explanation above and leave it at that. I won’t.

Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead:

0 1 2 3 4
9 8 7 6 5
By counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. However, our formula will look a bit different.

Notice that each column has a sum of n (not n+1, like before), since 0 and 9 are grouped. And instead of having exactly n items in 2 rows (for n/2 pairs total), we have n + 1 items in 2 rows (for (n + 1)/2 pairs total). If you plug these numbers in you get:

displaystyle{text{Number of pairs * Sum of each pair} = (frac{n + 1}{2})(n) = frac{n(n+1)}{2}}

which is the same formula as before. It always bugged me that the same formula worked for both odd and even numbers – won’t you get a fraction? Yep, you get the same formula, but for different reasons.

Technique 2: Use Two Rows
The above method works, but you handle odd and even numbers differently. Isn’t there a better way? Yes.

Instead of looping the numbers around, let’s write them in two rows:

1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1
Notice that we have 10 pairs, and each pair adds up to 10+1.

The total of all the numbers above is

displaystyle{text{Total = pairs * size of each pair} = n(n + 1)}

But we only want the sum of one row, not both. So we divide the formula above by 2 and get:

displaystyle{frac{n(n + 1)}{2}}

Now this is cool (as cool as rows of numbers can be). It works for an odd or even number of items the same!

Asked In Bank Exam KICK (6 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution