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Question
Find the sum 3 + 11 + 19 + ... + 803
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Solution
In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum of n terms of an A.P.,
`S_n = n/2[2a + (n -1)d]`
Where; a = first term for the given A.P.
d = common difference of the given A.P.
n = number of terms
3 + 11 + 19 + ... + 803
Common difference of the A.P. (d) = `a_2 - a_1`
= 19 - 11
= 8
So here,
First term (a) = 3
Last term (l) = 803
Common difference (d) = 8
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,
Further simplifying,
803 = 3 + (n -1)8
803 = 3 + 8n - 8
803 + 5 = 8n
808 = 8n
`n = 808/8`
n = 101
Now, using the formula for the sum of n terms, we get
`S_n = 101/2[2(3) + (101 - 1)8]`
`= 101/2 [6 + (100)8]`
`= 101/2 (806)`
= 101(403)
= 40703
Therefore, the sum of the A.P is `S_n = 40703`
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