A watermelon vendor arranged the watermelons similar to shown in the adjoining picture. The number of watermelons in subsequent rows differ by 'd'. The bottommost row has 101 watermelons and the topmost row has 1 watermelon. There are 21 rows from bottom to top. Based on the above information, answer the following questions :
(i) Find the value of 'd'.
(ii) How many watermelons will be there in the 15th row from the bottom? (iii) (a) Find the total number of watermelons from bottom to top. OR
(iii) (b) If the number of watermelons in the nth row from top is equal to the number of watermelons in the nth row from bottom, find the value of n.
Show Hint
The middle row of any AP with an odd number of terms is always the average of the first and last terms: $(101+1)/2 = 51$.
Given:
Bottommost row (a₁) = 101 watermelons
Topmost row (a₂₁) = 1 watermelon
Total number of rows (n) = 21
This forms an Arithmetic Progression (A.P.)
1) Find the value of ‘d’:
Formula: aₙ = a₁ + (n − 1)d
1 = 101 + (21 − 1)d
1 = 101 + 20d
20d = 1 − 101
20d = −100
d = −100 / 20
d = −5
Answer (i): d = −5
2) Watermelons in the 15th row from bottom:
Formula: a₁₅ = a₁ + (15 − 1)d
= 101 + 14(−5)
= 101 − 70
= 31
Answer (ii): 31 watermelons
3) (a) Total number of watermelons:
Formula: Sₙ = n/2 [a₁ + aₙ]
= 21/2 (101 + 1)
= 21/2 × 102
= 21 × 51
= 1071
Answer (iii)(a): 1071 watermelons
OR
3) (b) If number in nth row from top equals nth row from bottom:
Since total rows = 21
Middle row number = (21 + 1) / 2
= 22 / 2
= 11
Answer (iii)(b): n = 11
Conclusion:
Common difference is −5, 15th row has 31 watermelons, total watermelons are 1071, and the middle row where both sides are equal is the 11th row.
