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प्रश्न
Find the sum \[7 + 10\frac{1}{2} + 14 + . . . + 84\]
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उत्तर
(v) \[7 + 10\frac{1}{2} + 14 + . . . + 84\]
Common difference of the A.P is
(d) =
`=10 1/2-7`
`=21/2 - 7`
`=(21-14)/2`
`=7/2`
So here,
First term (a) = 7
Last term (l) = 84
Common difference (d) = `7/2`
So, here the first step is to find the total number of terms. Let us take the number of terms as n.
Now, as we know,
`a_n = a + (n-1) d`
So, for the last term,
`84 = 7 + (n-1) 7/2`
`84 = 7 + (7n)/2 - 7/2`
`84 = (14-7)/2 + (7n)/2`
84 (2) = 7 + 7n
Further solving for n,
7n = 168 - 7
`n = 161/7`
n = 23
Now, using the formula for the sum of n terms, we get
`S_n = 23/2 [2(7) + (23-1) 7/2]`
` = 23/2 [14+(22)7/2]`
`=23/2(14+77) `
`= 23/2 (91)`
On further simplification, we get,
`S_n = 2093/2`
Therefore, the sum of the A.P is `S_n = 2093/2`
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