- 14-1.1: Ordinary annuity $1,000 5% Annually 8
- 14-1.2: Ordinary annuity $ 500 4% Semiannually 4
- 14-1.3: Ordinary annuity $2,000 8% Quarterly 3
- 14-1.4: Annuity due $3,000 6% Semiannually 3
- 14-1.5: Annuity due $5,000 3% Annually 4
- 14-1.6: Annuity due $ 800 7% Annually 5
- 14-1.7: Manually find the future value of an ordinary annuity of $300 paid ...
- 14-1.8: Manually find the future value of an annuity due of $500 paid semia...
- 14-1.9: Find the future value of an ordinary annuity of $3,000 annually aft...
- 14-1.10: Len and Sharron Smith are saving money for their daughter Heather t...
- 14-1.11: Harry Taylor plans to pay an ordinary annuity of $5,000 annually fo...
- 14-1.12: Scott Martin is planning to establish a retirement annuity. He is c...
- 14-1.13: Pat Lechleiter pays an ordinary annuity of $2,500 quarterly at 8% a...
- 14-1.14: Find the future value of an ordinary annuity of $6,500 semiannually...
- 14-1.15: Latanya Brown established an ordinary annuity of $1,000 annually at...
- 14-1.16: You invest in an ordinary annuity of $500 annually at 8% annual int...
- 14-1.17: You invest in an ordinary annuity of $2,000 annually at 8% annual i...
- 14-1.18: Make a chart comparing your results for Exercises 16 and 17. Use th...
- 14-1.19: Find the future value of an annuity due of $12,000 annually for thr...
- 14-1.20: Bernard McGhee has decided to establish an annuity due of $2,500 an...
- 14-1.21: Find the future value of an annuity due of $7,800 annually for two ...
- 14-1.22: Find the future value of an annuity due of $400 annually for two ye...
- 14-1.23: Find the future value of a quarterly annuity due of $4,400 for thre...
- 14-1.24: Find the future value of an annuity due of $750 semiannually for fo...
- 14-1.25: Which annuity earns more interest: an annuity due of $300 quarterly...
- 14-1.26: You have carefully examined your budget and determined that you can...
- 14-1.27: June Watson is contributing $3,000 each year to a Roth IRA. The IRA...
- 14-1.28: Marvin Murphy contributes $400 per month to a payroll deduction 401...
Solutions for Chapter 14-1: FUTURE VALUE OF AN ANNUITY
Full solutions for Business Math, | 9th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Solvable system Ax = b.
The right side b is in the column space of A.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.