Question

In a mango groove the trees are planted in horizontal rows.
There are 6 trees more in each horizontal row. Altogether
there are 720 trees. Find the number of trees in horizontal rows.​

Answer 1

This is a problem is AP, where the sum is 720 and d= 6. Both a and n are unknown. Let n = number of rows and a = number of trees in the first row.

Sn = (n/2)[2a+(n-1)d], or

720 = (n/2)[2a+(n-1)6], or

720 = n[a+(n-1)3], or

720 = an +3n(n-1) = 3n^2 +(a-3)n

3n^2 -(3-a)n - 720 = 0 …(1)

If a = 3, n = (720/3)^0.5 = 15.5. Rejected

If a = 6, (1) becomes 3n^2 -(3–6)n - 720 = 0 or

3n^2 +3n - 720 = 0 or

n^2+n-240=0

(n+16)(n-15) = 0

Or n = 15 and as a negative value of n = -16 is inadmissible.

So there are 15 rows. The first row has 6 and in each subsequent row 6 more trees are added.

Check: Sn =(15/2)[2*6 +(15–1)*6]

= (15/2)[12+14*6]

= (15/2)[12+84]

= (15/2)*96

= 720. Correct.