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Question 1 of 4
Evaluate
16∑n=5(5n−7)16∑n=5(5n−7)
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First, substitute the values x=5 to 16x=5 to 16 to the expression
16∑x=5(5n−7)16∑x=5(5n−7) |
== |
(5((5(55)-7)+(5()−7)+(5(66)-7)+(5()−7)+(5(77)-7)…(5()−7)…(5(1616)-7))−7) |
|
== |
1818+23+28…+23+28…7373 |
The first term (aa) is 1818 and the last term (ll) is 7373.
Next, find the number of terms by subtracting the lower value of xx from its highest value then adding 11.
nn |
== |
(16-5)+1(16−5)+1 |
|
== |
1212 |
Finally, substitute the known values to the formula
nn |
== |
1212 |
aa |
== |
1818 |
ll |
== |
7373 |
SnSn |
== |
n2[a+l]n2[a+l] |
|
S12S12 |
== |
122[18+73]122[18+73] |
Substitute known values |
|
|
== |
6[91]6[91] |
Evaluate |
|
== |
546546 |
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Question 2 of 4
Evaluate
10∑x=43n10∑x=43n
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First, substitute the values x=4 to 10x=4 to 10 to the expression
10∑x=43n10∑x=43n |
== |
34+35+36…31034+35+36…310 |
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== |
8181+243+729…59 049+243+729…59 049 |
The first term (aa) is 8181.
Next, find the common ratio (rr) by dividing two consecutive terms.
rr |
== |
35÷3435÷34 |
|
== |
35-435−4 |
|
== |
33 |
Next, find the number of terms by subtracting the lower value of xx from its highest value then adding 11.
nn |
== |
(10-4)+1(10−4)+1 |
|
= |
7 |
Finally, substitute the known values to the formula
Sn |
= |
a(rn−1)r−1 |
|
S7 |
= |
81(37−1)3−1 |
Substitute known values |
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|
= |
81(2186)2 |
Evaluate |
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|
= |
177 0662 |
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|
= |
88 533 |
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Question 3 of 4
Find the limiting sum
∞∑r=2(45)r−1
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Limiting Sum Formula
S∞=a1−r
where -1<r<1
Common Ratio Formula
r=U2U1=U3U2
First, substitute the values r=2.. to the expression
∞∑r=2(45)r−1 |
= |
(45)2−1+(45)3−1(45)4−1… |
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|
= |
45+1625+64125… |
The first term (a) is 45.
Next, find the common ratio (r) by dividing two consecutive terms.
r |
= |
1625÷45 |
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|
= |
80100 |
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|
= |
45 |
Simplify |
Finally, substitute the known values to the limiting sum formula
S∞ |
= |
a1−r |
|
S∞ |
= |
451−45 |
Substitute known values |
|
|
= |
4515 |
Evaluate |
|
|
= |
205 |
|
|
= |
4 |
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Question 4 of 4
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A sigma notation can be used to get the sum of values without having a common difference or ratio.
Substitute the values r=3 to 6 to the expression and evaluate
6∑r=3(−1)rr2 |
= |
[(−1)3⋅32]+[(−1)4⋅42]+[(−1)5⋅52]+[(−1)6⋅62] |
|
= |
[(-1)⋅9]+[1⋅16]+[(-1)⋅25]+[1⋅36] |
Evaluate |
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= |
-9+16-25+36 |
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= |
18 |