M60(Precalculus) Evaluate the factorial expression. 9! ) 7!! ch practice test ) Write out the first five terms of the sequence. ) {sn} = (-)n - n + n - ) The given pattern continues. Write down the nth term of the sequence {an} suggested by the pattern. 3) 4, 0, 6,, 8,... 3) 4) 3, -6, 9, -,... 4) ) 3, 4, 3,,... ) 4 6 6) During a five-year period, a company doubles its profits each year. If the profits at the end of the fifth year are $60,000, then what are the profits for each of the first four years? 6) The sequence is defined recursively. Write the first four terms. 7) a = ; an = an- - 3 7) 8) a =, a = ; an = an- - 3an- 8)
9) Given that a = -4, a = -4 and an+ = an+ - 4an, what is the fifth term of this recursively defined sequence? 9) 0) Jake bought a truck by taking out a loan for $6,00 at 0.% interest per month. Jake's regular monthly payment is $67, but he decides to pay an extra $7 toward the balance each month. His balance each month, after making his payment, is given by the recursively defined sequence B0 = $6,00 Bn =.00Bn-- 64 Determine Jake's balance after making the first payment. That is, determine B. 0) Write out the sum. ) n k = (k + 3) ) n ) - k = 0 k + ) Express the sum using summation notation. 3) + 8 + 8 +... + 7 3) 4) 4-6 + 64 -... + (-) ( + ) 4 4) ) 4 + 8 3 + 4 +... + 3 9 )
Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are given. 6) a = 9; d = -6 6) an =?; a6 =? Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. 7) 6th term is -0; th term is -46 7) Find the indicated term using the given information. 8) a 48 = - 37, a 6 = - 7 ; a 3 8) Find the sum. 9) (-6) + (-) + 4 + 9 +... + 39 9) 0) 4 n = (n - 3) 0) ) A theater has rows with 30 seats in the first row, 34 in the second row, 38 in the third row, and so forth. How many seats are in the theater? ) ) Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $00 per year. How many years will it be before you've made a total (or aggregate) salary of $,0,000? ) 3) A local civic theater has seats in the first row and rows in all. Each successive row contains 3 additional seats. How many seats are in the civic theater? 3) Find the nth term {an} of the geometric sequence. When given, r is the common ratio. 4) 4,,,,... 4) ) 3, - 3, 3 4, - 3 8, 3,... ) 6 6) A new piece of equipment cost a company $49,000. Each year, for tax purposes, the company depreciates the value by %.What value should the company give the equipment after 8 years? 6) 7) A football player signs a contract with a starting salary of $80,000 per year and an annual increase of 6.% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the seventh year? 7) 3
8) Jennifer takes a job with a starting salary of $43,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the seventh year? 8) Find the sum. 9) 3 k = 3 k+ 9) Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. 30) + + + 3 +... + 3 30) 3) Initially, a pendulum swings through an arc of 3 feet. On each successive swing, the length of the arc is 0.8 of the previous length. After 0 swings, what total length will the pendulum have swung (to the nearest tenth of a foot)? 3) Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. 3) - 4 + 6 -... 3) 33) + 6 + 3 + 3 +... 33) 34) 4 3 k= k- 34) Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. 3) 3 + 8 + 3 +... + (n - ) = n (n + ) 3) 36) n - n + is divisible by 36) 37) 9 + 9 + 3 9 +... + 9n = 9n(n + ) 37) Expand the expression using the Binomial Theorem. 38) (x - y)4 38) Use the Binomial Theorem to find the indicated coefficient or term. 39) The coefficient of x 4 in the expansion of (3x + 4) 6 39) 40) The th term in the expansion of (x - y)4 40) 4
Answer Key Testname: CHPRAC ) 36 ) s=, s= -, s3= 4, s 4= - 7, s = 3 3) an = ( 3 n - ) 4) an = (-) n + 3n ) an = n(n + ) 6) $0,000, $0,000, $40,000, $80,000 7) a =, a =, a3 = -, a4 = - 8) a =, a =, a3 = -3, a4 = 44 9) a = -0 0) $,94. ) 6 + + 36 +... + (n + 3) ) + 4 + 8 +... + n 3) 6 k k = 4) (-) (k+) 4 k k = ) 8 (4k) k + k = 6) an = 98-6n; a6 = 6 7) a = 0, d = -4, an = an- - 4 8) - 9) 6 0) 040 ) 84 seats ) 0 years 3) 09 seats 4) an = 4 n- ) an = 3 - n- 6) $4906 7) $,96,497 8) $4,409 9) 76 8 30) 376.6 3) approximately 3.4 feet
Answer Key Testname: CHPRAC 3) Converges; 4 33) Converges; 4 34) Converges; 3) First we show that the statement is true when n =. For n =, we get 3 = () (() + ) = 3. This is a true statement and Condition I is satisfied. Next, we assume the statement holds for some k. That is, 3 + 8 + 3 +... + (k - ) = k (k + ) is true for some positive integer k. We need to show that the statement holds for k +. That is, we need to show that 3 + 8 + 3 +... + ((k + ) - ) = k + ((k + ) + ). So we assume that 3 + 8 + 3 +... + (k - ) = k (k + ) is true and add the next term, (k + ) -, to both sides of the equation. 3 + 8 + 3 +... + (k - ) + (k + ) - = k (k + ) + (k + ) - = [k(k + ) + 0(k + ) - 4] = (k + k + 0k + 0-4) = (k + k + 6) = (k + )(k + 6) = k + (k + + ) = k + ((k + ) + ) Condition II is satisfied. As a result, the statement is true for all natural numbers n. 6
Answer Key Testname: CHPRAC 36) First, we show that the statement is true when n =. For n =, n - n + = () - () + =. This is a true statement and Condition I is satisfied. Next, we assume the statement holds for some k. That is, k - k + is divisible by is true for some positive integer k. We need to show that the statement holds for k +. That is, we need to show that (k + ) - (k + ) + is divisible by. So we assume k - k + is divisible by and look at the expression for n = k +. (k + ) - (k + ) + = k + k + - k - + = (k - k + ) + k Since k - k + is divisible by, then k - k + = m for some integer m. Hence, (k + ) - (k + ) + = (k - k + ) + k = m + k = (m + k). Condition II is satisfied. As a result, the statement is true for all natural numbers n. 37) First we show that the statement is true when n =. 9()( + ) For n =, we get 9 = = 9. This is a true statement and Condition I is satisfied. Next, we assume the statement holds for some k. That is, 9k(k + ) 9 + 9 + 3 9 +... + 9k = is true for some positive integer k. We need to show that the statement holds for k +. That is, we need to show that 9(k + )(k + ) 9 + 9 + 3 9 +... + 9(k + ) =. So we assume that 9 + 9 + 3 9 +... + 9k = equation. 9 + 9 + 3 9 +... + 9k + 9(k + ) = = = 9k(k + ) 9k(k + ) + 9(k + ) 9[k(k + ) + (k + )] 9(k + )(k + ) Condition II is satisfied. As a result, the statement is true for all natural numbers n. 38) x8-0x6y + 0x4y - 00x y 3 + 6y 4 39) 9,440 40) -,963,776x3y is true and add the next term, 9(k + ), to both sides of the 7