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Symbols and Formulas: Arithmetic Sequence
$AS_n$ = $nth$ term of an Arithmetic Sequence
$a$ = first term
$p$ = last term
$d$ = common difference
$n$ = number of terms
$SAS_n$ = sum of the first $n$ terms of an Arithmetic Sequence
$
(1.)\:\: AS_n = a + d(n - 1) \\[5ex]
(2.)\:\: AS_n = vn + w \:\:where\:\: v = d \:\:and\:\: w = a - d \\[5ex]
(3.)\:\: p = a + d(n - 1) \\[5ex]
(4.)\:\: SAS_n = \dfrac{n}{2}(a + AS_n) \\[7ex]
(5.)\:\: SAS_n = \dfrac{n}{2}(a + p) \\[7ex]
(6.)\:\: SAS_n = \dfrac{n}{2}[2a + d(n - 1)] \\[7ex]
(7.)\:\: n = \dfrac{2 * SAS_n}{a + p} \\[7ex]
(8.)\:\: n = \dfrac{p - a + d}{d} \\[7ex]
(9.)\:\: n = \dfrac{-(2a - d) \pm \sqrt{(2a - d)^2 + 8d*SAS_n}}{2d} \\[7ex]
(10.)\;\; d = \dfrac{(p - a)(p + a)}{2 * SAS_n - p - a}
$
Symbols and Formulas: Geometric Sequence
$GS_n$ = $nth$ term of a Geometric Sequence
$a$ = first term
$p$ = last term
$r$ = common ratio
$n$ = number of terms
$SGS_n$ = sum of the first $n$ terms of a Geometric Sequence
$S_{\infty}$ = sum to infinity of a Geometric Sequence
(2.) The 3rd and 6th terms of a geometric progression (G.P) are 2 and 54 respectively.
Find the:
(a.) common ratio
(b.) first term
(c.) sum of the first ten terms, correct to the nearest whole number.
(3.) The sum of the second and third terms of a Geometric Progression (G.P) is 48
If the sum of the third and fourth terms is 144, find the first term of the progression.
(5.) The sum of the first twelve terms of an Arithmetic Progression is 168
If the third term is 7, find the values of the common difference and the first term.
(7.) The $nth$ term of an Arithmetic Progression (A.P) is given by $U_n = n^2\log 5$
Find the sum of the first n terms of the Arithmetic Progression in terms of $\log 5$
(8.) If the sixth term of an Arithmetic Progression (A.P) is 37 and the sum of the first six terms is 147,
find the:
(a.) first term
(b.) sum of the first fifteen terms
(9.) The sum of the first n terms of a sequence is given by $S_n = \dfrac{5n^2}{2} + \dfrac{5n}{2}$
Write down the first four terms of the sequence and find an exprsssion for the nth term.
We can solve this question in at least two ways.
Use any method you prefer.
We are given the sum of the first $n$ terms
However, we do not know the kind of sequence
So, let us try to determine the kind of sequence by testing some numbers
Let us find the sum of the first one term; sum of the first two terms; sum of the first three terms;
sum of the first four terms
Then, we can find the four terms of the sequence
And determine the fifth term and the sum of the first five terms
And verify that sum by finding the sum of the first five terms.
(11.) (a.) If $3, x, y, 18$ are in Arithmetic Progression (A.P), find the values of x and y
(b.) (i.) The sum of the second and third terms of a geometric progression is six times the fourth term.
Find the two possible values of the common ratio.
(ii.) If the second term is 8 and the common ratio is positive, find the first six terms.
$
(a.) \\[3ex]
AP:\:\: 3, x, y, 18 \\[3ex]
d = x - 3 \\[3ex]
d = y - x \\[3ex]
d = 18 - y \\[3ex]
d = d \\[3ex]
\implies \\[3ex]
x - 3 = y - x \\[3ex]
2x - 3 = y \\[3ex]
y = 2x - 3...eqn.(1) \\[3ex]
\implies \\[3ex]
y - x = 18 - y \\[3ex]
2y - x = 18 \\[3ex]
Substitute\:\: (2x - 3)\:\:for\:\:y \\[3ex]
2(2x - 3) - x = 18 \\[3ex]
4x - 6 - x = 18 \\[3ex]
3x - 6 = 18 \\[3ex]
3x = 18 + 6 \\[3ex]
3x = 24 \\[3ex]
x = \dfrac{24}{3} \\[5ex]
x = 8 \\[3ex]
Substitute\:\:x = 8\:\:in\:\:eqn.(1) \\[3ex]
y = 2x - 3 \\[3ex]
y = 2(8) - 3 \\[3ex]
y = 16 - 3 \\[3ex]
y = 13 \\[3ex]
AP:\:\: 3, 8, 13, 18 \\[5ex]
(b.) \\[3ex]
(i.) \\[3ex]
GP_n = ar^{n - 1} \\[3ex]
GP_2 = ar \\[3ex]
GP_3 = ar^2 \\[3ex]
GP_4 = ar^3 \\[3ex]
GP_2 + GP_3 = 6 * GP_4 \\[3ex]
ar + ar^2 = 6(ar^3) \\[3ex]
a(r + r^2) = 6ar^3 \\[3ex]
Divide\:\:both\:\:sides\:\:by\:\:a \\[3ex]
r + r^2 = 6r^3 \\[3ex]
0 = 6r^3 - r^2 - r \\[3ex]
6r^3 - r^2 - r = 0 \\[3ex]
r(6r^2 - r - 1) = 0 \\[3ex]
Factor\:\: 6r^2 - r - 1 \\[3ex]
6r^2 + 2r - 3r - 1 \\[3ex]
2r(3r + 1) - 1(3r + 1) \\[3ex]
(3r + 1)(2r - 1) \\[3ex]
\implies \\[3ex]
r(3r + 1)(2r - 1) = 0 \\[3ex]
r = 0 \:\:OR \\[3ex]
3r + 1 = 0 \\[3ex]
3r = 0 - 1 \\[3ex]
3r = -1 \\[3ex]
r = -\dfrac{1}{3} \:\:OR \\[5ex]
2r - 1 = 0 \\[3ex]
2r = 0 + 1 \\[3ex]
2r = 1 \\[3ex]
r = \dfrac{1}{2} \\[5ex]
r = -\dfrac{1}{3},0,\dfrac{1}{2} \\[5ex]
r \ne 0 \\[3ex]
\therefore r = -\dfrac{1}{3},\dfrac{1}{2} \\[5ex]
(ii.) \\[3ex]
r \:\:is\:\:positive \\[3ex]
\implies \\[3ex]
r = \dfrac{1}{2} \\[5ex]
GP_2 = ar = 8 \\[3ex]
a = 8 \div r \\[3ex]
a = 8 \div \dfrac{1}{2} \\[5ex]
a = 8 * \dfrac{2}{1} \\[5ex]
a = 16 \\[3ex]
GP_3 = ar^2 = ar * r = GP_2 * r = 8 * \dfrac{1}{2} = 4 \\[5ex]
GP_4 = GP_3 * r = 4 * \dfrac{1}{2} = 2 \\[5ex]
GP_5 = GP_4 * r = 2 * \dfrac{1}{2} = 1 \\[5ex]
GP_6 = GP_5 * r = 1 * \dfrac{1}{2} = \dfrac{1}{2} \\[5ex]
GP:\:\: 16, 8, 4, 2, 1, \dfrac{1}{2}
$
