write the sigma notation.
5^2+6^2+7^2+...+14^2
You could use capatil E as a sigma.
Please help!!
5 = 1 + 4
6 = 2 + 4
7 = 3 + 4
...
14 = 10 + 4
5 ^ 2 + 6 ^ 2 + 7 ^ 2 + ... + 14 ^ 2 =
( 1 + 4 ) ^ 2 + ( 2 + 4 ) ^ 2 + ( 3 + 4 ) ^ 2 + ... + ( 10 + 4 )^ 2 =
E ( k + 4 ) ^ 2 from n = 1 to n = 10
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( k + 4 ) ^ 2 = k ^ + 2 * k * 4 + 4 ^ 2 =
k ^ 2 + 8 k + 16
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E ( k + 4 ) ^ 2 from n = 1 to n = 10 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + E 16 from n = 1 to n = 10 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 10 * 16 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 160
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E k ^ 2 from n = 1 to n = 10
= n * ( n + 1 ) * ( 2 n + 1 ) / 6 =
10 * ( 10 + 1 ) * ( 2 * 10 + 1 ) / 6 =
10 * 11 * 21 / 6 = 2310 / 6 = 385
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E k from n = 1 to n = 10
= n * ( n + 1 ) / 2 = 10 * 11 / 2 = 110 / 2 = 55
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E ( k + 4 ) ^ 2 from n = 1 to n = 10
= E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 160
= 385 + 8 * 55 + 160 = 385 + 440 + 160 = 985
To write the given expression in sigma notation using the uppercase sigma symbol (∑), we need to specify the starting index and the final index, and define the expression inside the sigma that needs to be evaluated for each value within that range.
In this case, the given expression:
5^2 + 6^2 + 7^2 + ... + 14^2
can be rewritten using sigma notation as:
∑(n=5 to 14) n^2
Here, the value of "n" takes on all the integers starting from 5 and going up to 14. For each value of "n," we calculate n^2 and then sum up all these squared values.
Hope this helps!