00:01
In the given question we are told that a farmer has to plant seeds in a triangular field and he knows that the middle row, row 26 needs 4 ,025 seeds and the last row needs 7 ,525 seeds.
00:22
So let's write this down.
00:24
The 26th row needs 4 ,000.
00:30
25 seeds and we are told that this is the middle row and the last row the last row is said to need 7 ,525 seeds.
00:54
So this is what the farmer knows and we are asked if the number of seeds planted in each row follows arithmetic sequence follows an arithmetic sequence an arithmetic sequence then arithmetic sequence then how many total seeds does he need so this is what we are asked we should find the total seeds the farmer needs so then we are told that this is an arithmetic sequence right so in an arithmetic sequence if you assume that the sequence is a 1 a 2 a 3 etc up to a n then if if this is an arithmetic sequence in an arithmetic sequence in an arithmetic sequence arithmetic sequence there is a common difference between these terms which means we would get a 2 by just adding some common difference that is some constant let's say d to a1 and in order to obtain the third term that is a3 we just have to add the same common difference to a2 and this goes on for all these terms so d is called the common difference and the general formula for a .n that is the nth term in an arithmetic sequence is given by a plus n minus 1 times d so this is the formula that we need to know where d is the common difference the common difference and a n is the nth term the nth term a is the first term of the sequence first term and these are what we need to know about the given formula so over here we are told that this the middle row is the row 26 right so we are told the middle row the middle row is row 26 which implies that there is a total there is a total number there is a there is an odd total number of seeds right there is an odd total number of seeds now the reason why we would say this is that when we look at the middle row of let's say let's take an an odd number of rows that is three rows right so if there are three rows 1 2 and 3 the middle row would be the second row, right? and if it is 1, 2, 3, 4, if it is an even row, both these rows would come in as the middle rows.
04:31
If it is an even number of rows, not number of seeds, even number of rows.
04:38
So since the middle row over here is given as an even number number, the number.
04:45
We could assume that the total number of rows is odd numbered which means we can take the middle part of an odd number of rows as n plus 1 by 2 n plus 1 by 2 is equal to 26 from which we can write n plus 1 is equal to 52 or n is equal to 52 minus 1 which is 51 so 51 is the total number of rows total number of rows so now we know that 51 is the last row right so for the 26th row we can write according to the formula that we have just mentioned the 26th row can be taken as a plus times d that is n minus 1 times d and the 51st row can be taken as a plus 50 times d and now we know that these two rows that is the 26th row needs 4 ,025 4 ,025 seeds and the 50 first row needs 7 ,525 seeds...
