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Step-by-step explanation:
It is given that the longer side of trapezoid needs to start with a row of 97 bricks and each row must be decreased by 2 on each end and the construction should stop at 25th row.
Let us write the number of bricks in each row as an arithmetic sequence as follows:
97,93,89,
Now, we need to find the number of bricks needed to buy and for that we have to write the above sequence as series that is
97+93+89+
The first term of the series is a
1
=97, the second term is a
2
=93 and so on.
We find the common difference d by subtracting the first term from the second term as shown below:
d=a
2
−a
1
=93−97=−4
We know that the sum of an arithmetic series with first term a and common difference d is S
n
=
2
n
[2a+(n−1)d]
Thus, substitute a=97,d=−4 and n=25 in S
n
=
2
n
[2a+(n−1)d] as follows:
S
25
=
2
25
[(2×97)+(25−1)(−4)]=
2
25
[194+(24×−4)]=
2
25
[194−96]=
2
25
×98=25×49=1225
Hence, the gardener needs to buy 1225 bricks.