7. What are the values of these sums?4a) Σα+1)b) Σ-2))j=0k=1108c) E 3d) E (2j+1 – 2i)i=1j=0 SummationsNext, we consider the addition of the terms of a sequence. For this we introduce summationnotation. We begin by describing the notation used to express the sum of the termsam. dm+l.... dnfrom the sequence (a,). We use the notationEasisa@jor(read as the sum from j = m to j =n of a ) to representam + am+1 +...+ an.Here, the variable j is called the index of summation, and the choice of the letter j as thevariable is arbitrary: that is, we could have used any other letter, such as i or k. Or, in notation,Here, the index of summation runs through all integers starting with its lower limit m and endingwith its upper limit n. A large uppercase Greek letter sigma, E, is used to denote summation.The usual laws for arithmetic apply to summations. For example, when a and b are realnumbers, we have E-(ax; + by) = a E- j +bE-Yj, where x, X2, ..., X, andy1. 2..... Ya are real numbers. (We do not present a formal proof of this identity here. Such aproof can be constructed using mathematical induction, a proof method we introduce in Chap-ter 5. The proof also uses the commutative and associative laws for addition and the distributivelaw of multiplication over addition.)We give some examples of summation notation.EXAMPLE 17 Use summation notation to express the sum of the first 100 terms of the sequence (a,}, wherea = 1/j for j = 1, 2, 3. ....ExtraExamplesSolution: The lower limit for the index of summation is 1, and the upper limit is 100. We writethis sum as100EXAMPLE 18 What is the value of E-?Solution: We haveE-+2+ 3²+ 4² + 5²=I+4+9+ 16+ 25- 55.EXAMPLE 19 What is the value of -1)?Solution: We have(-1)* = (-1)4 + (-1) +(-1)* +(-1)7 +(-1)=1+(-1) +1+(-1) +1Sometimes it is useful to shift the index of summation in a sum. This is often done whentwo sums need to be added but their indices of summation do not match. When shifting an indexof summation, it is important to make the appropriate changes in the corresponding summand.This is illustrated by Example 20.EXAMPLE 20 Suppose we have the sumExtrabut want the index of summation to run between 0 and 4 rather than from I to 5. To do this,Examples we let k = j -1. Then the new summation index runs from 0 (because k = 1-0=0 whenj = 1) to 4 (because k = 5-1= 4 when j= 5), and the term j becomes (k +.Hence,EP =Ek + 1).It is easily checked that both sums are I+4+9+ 16 + 25 - 55.Sums of terms of geometric progressions commonly arise (such sums are called geometricseries). Theorem I gives us a formula for the sum of terms of a geometrie progression.

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