Which of the summation properties or formulas require the lower index to be 1? Check all that apply.
[tex]\sum_{i=?}^n i^3=\left[\frac{n(n+1)}{2}\right]^2[/tex]
[tex]\sum_{i=?}^n\left(a_i \pm b_i\right)=\sum_{i=?}^n a_i \pm \sum_{i=?}^n b_i[/tex]
[tex]\sum_{i=?}^n i=\frac{n(n+1)}{2}[/tex]
[tex]\sum_{i=?}^n i^2=\frac{n(n+1)(2 n+1)}{6}[/tex]
[tex]\sum_{i=?}^n c a_i=c \sum_{i=?}^n a_i[/tex]
[tex]\sum_{i=?}^n c=c n[/tex]
Answer (2)
The formulas ∑ i = ? n i 3 = [ 2 n ( n + 1 ) ] 2 , ∑ i = ? n i = 2 n ( n + 1 ) , ∑ i = ? n i 2 = 6 n ( n + 1 ) ( 2 n + 1 ) , and ∑ i = ? n c = c n require the lower index to be 1.
The formula ∑ i = ? n ( a i ± b i ) = ∑ i = ? n a i ± ∑ i = ? n b i is a property of summation and holds true regardless of the lower index.
The formula ∑ i = ? n c a i = c ∑ i = ? n a i is a property of summation and holds true regardless of the lower index.
Therefore, the formulas that require the lower index to be 1 are i = ? ∑ n i 3 = [ 2 n ( n + 1 ) ] 2 , i = ? ∑ n i = 2 n ( n + 1 ) , i = ? ∑ n i 2 = 6 n ( n + 1 ) ( 2 n + 1 ) , i = ? ∑ n c = c n .
Explanation
Analyzing the Problem We need to determine which of the given summation formulas require the lower index to be 1. Let's analyze each formula individually.
Analyzing the first formula
∑ i = ? n i 3 = [ 2 n ( n + 1 ) ] 2
This formula is valid when the lower index is 1. If the lower index is 0, the formula is still valid because 0 3 = 0 . However, if the lower index is 2, the formula is no longer valid. For example, if n = 3 , ∑ i = 2 3 i 3 = 2 3 + 3 3 = 8 + 27 = 35 , but [ 2 3 ( 3 + 1 ) ] 2 − 1 3 = [ 2 12 ] 2 − 1 = 36 − 1 = 35 . This might seem correct, but the original formula [ 2 n ( n + 1 ) ] 2 is only valid when the lower index is 1. So, this formula requires the lower index to be 1.
Analyzing the second formula
∑ i = ? n ( a i ± b i ) = ∑ i = ? n a i ± ∑ i = ? n b i
This is a property of summation and holds true regardless of the lower index. It's a general property of sums.
Analyzing the third formula
∑ i = ? n i = 2 n ( n + 1 )
This formula is valid when the lower index is 1. If the lower index is 0, the formula is still valid because adding 0 doesn't change the sum. However, if the lower index is 2, the formula is no longer valid. For example, if n = 3 , ∑ i = 2 3 i = 2 + 3 = 5 , but 2 3 ( 3 + 1 ) − 1 = 2 12 − 1 = 6 − 1 = 5 . This might seem correct, but the original formula 2 n ( n + 1 ) is only valid when the lower index is 1. So, this formula requires the lower index to be 1.
Analyzing the fourth formula
∑ i = ? n i 2 = 6 n ( n + 1 ) ( 2 n + 1 )
This formula is valid when the lower index is 1. If the lower index is 0, the formula is still valid because 0 2 = 0 . However, if the lower index is 2, the formula is no longer valid. For example, if n = 3 , ∑ i = 2 3 i 2 = 2 2 + 3 2 = 4 + 9 = 13 , but 6 3 ( 3 + 1 ) ( 2 ( 3 ) + 1 ) − 1 2 = 6 3 ( 4 ) ( 7 ) − 1 = 6 84 − 1 = 14 − 1 = 13 . This might seem correct, but the original formula 6 n ( n + 1 ) ( 2 n + 1 ) is only valid when the lower index is 1. So, this formula requires the lower index to be 1.
Analyzing the fifth formula
∑ i = ? n c a i = c ∑ i = ? n a i
This is a property of summation and holds true regardless of the lower index. It's a general property of sums.
Analyzing the sixth formula
∑ i = ? n c = c n
This formula requires the lower index to be 1. If the lower index is 0, the formula is still valid. However, if the lower index is 2, the formula is no longer valid. The correct formula would be ∑ i = 1 n c = c n . So, this formula requires the lower index to be 1.
Final Answer Therefore, the summation formulas that require the lower index to be 1 are:
∑ i = ? n i 3 = [ 2 n ( n + 1 ) ] 2
∑ i = ? n i = 2 n ( n + 1 )
∑ i = ? n i 2 = 6 n ( n + 1 ) ( 2 n + 1 )
∑ i = ? n c = c n
Examples
Summation formulas are used in various fields, such as physics, engineering, and computer science. For example, when calculating the total distance traveled by an object with increasing velocity, we can use summation formulas to find the sum of the distances traveled in each time interval. These formulas help simplify complex calculations and provide efficient solutions to real-world problems. Understanding the conditions under which these formulas are valid is crucial for accurate results.
The summation formulas that require the lower index to be 1 include ∑ i = 1 n i 3 = [ 2 n ( n + 1 ) ] 2 , ∑ i = 1 n i = 2 n ( n + 1 ) , ∑ i = 1 n i 2 = 6 n ( n + 1 ) ( 2 n + 1 ) , and ∑ i = 1 n c = c n . Other properties, such as the sum of sequences and multiplication by a constant, do not require the lower index to be 1.
;
