SECTION 12.1 Sequences 861 4 Find the Sum of a Sequence Algebraically and Using a Graphing Utility The following theorem lists some properties of summation notation.These properties are useful for adding the terms of a sequence. THEOREM Summation Notation Properties If { } an and { } bn are two sequences and c is a real number, then ca ca ca ca c a a a c a k n k n n k n k 1 1 2 1 2 1 ∑ ∑ ( ) ( ) = + + + = + + + = = = (1) ∑ ∑ ∑ ( ) + = + = = = a b a b k n k k k n k k n k 1 1 1 (2) ∑ ∑ ∑ ( ) − = − = = = a b a b k n k k k n k k n k 1 1 1 (3) ∑ ∑ ∑ = − < < = + = = a a a j n where 0 k j n k k n k k j k 1 1 1 (4) The proof of property (1) follows from the distributive property of real numbers. The proofs of properties (2) and (3) are based on the commutative and associative properties of real numbers. Property (4) states that the sum from +j 1 to n equals the sum from 1 to n minus the sum from 1 to j. This property is helpful when the index of summation begins at a number larger than 1. The next theorem provides some formulas for finding the sum of certain sequences. THEOREM Formulas for Sums of the First n Terms of a Sequence ∑ = + + + = = c c c c cn c is a real number k n 1 nterms (5) k n n n 1 2 3 1 2 k n 1 ∑ ( ) = + + + + = + = (6) k n n n n 1 2 3 1 2 1 6 k n 1 2 2 2 2 2 ∑ ( )( ) = + + + + = + + = (7) k n n n 1 2 3 1 2 k n 1 3 3 3 3 3 2 ∑ ( ) = + + + + = ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = (8) The proof of formula (5) follows from the definition of summation notation.You are asked to prove formula (6) in Problem 100. The proofs of formulas (7) and (8) require mathematical induction, which is discussed in Section 12.5. Notice the difference between formulas (5) and (6). In (5), the constant c is being summed from 1 to n, while in (6) the index of summation k is being summed from 1 to n. Finding the Sum of a Sequence Find the sum of each sequence. (a) k3 k 1 5 ∑( ) = (b) ∑( ) + = k 1 k 1 10 3 (c) ∑( ) − + = k k7 2 k 1 24 2 (d) ∑( ) = k4 k 6 20 2 EXAMPLE 9 (continued)
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