860 CHAPTER 12 Sequences; Induction; the Binomial Theorem Rather than writing down all these terms, we use summation notation to express the sum more concisely: ∑ + + + + = = a a a a a n k n k 1 2 3 1 The symbol Σ (the Greek letter sigma, which is an S in our alphabet) is simply an instruction to sum, or add up, the terms. The integer k is called the index of the sum; it tells where to start the sum and where to end it. The expression ∑ = a k n k 1 is an instruction to add the terms ak of the sequence { } an starting with = k 1 and ending with = k n. The expression is read as “the sum of ak from = k 1 to = k n.” Expanding Summation Notation Expand each sum. (a) ∑ = k 1 k n 1 (b) ∑ = k ! k n 1 EXAMPLE 7 Solution (a) ∑ = + + + + = k n 1 1 1 2 1 3 1 k n 1 (b) k n ! 1! 2! ! k n 1 ∑ = + + + = Now Work PROBLEM 53 Expressing a Sum Using Summation Notation Express each sum using summation notation. (a) 1 2 3 9 2 2 2 2 + + + + (b) 1 1 2 1 4 1 8 1 2n 1 + + + + + − Solution EXAMPLE 8 (a) The sum + + + + 1 2 3 9 2 2 2 2 has 9 terms, each of the form k ,2 starting at = k 1 and ending at = k 9: k 1 2 3 9 k 2 2 2 2 1 9 2 ∑ + + + + = = (b) The sum 1 1 2 1 4 1 8 1 2n 1 + + + + + − has n terms, each of the form − 1 2 , k 1 starting at = k 1 and ending at = k n: 1 1 2 1 4 1 8 1 2 1 2 n k n k 1 1 1 ∑ + + + + + = − = − Now Work PROBLEM 63 The index of summation need not begin at 1, nor end at n; for example, the sum in Example 8(b) could also be expressed as ∑ = + + + + = − − 1 2 1 1 2 1 4 1 2 k n k n 0 1 1 Letters other than k are also used as the index. For example, j i !and ! j n i n 1 1 ∑ ∑ = = both represent the same sum given in Example 7(b).
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