WISKUNDE
GRAAD 12
NOG OEFENINGE
Meetkundige reekse : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Geometric series : answers.
T2
8
T2
20
5
1.1
a = T1; = 2 ; r = ── = ── = 2
2.1
a = T1; = 16 ; r = ── = ── = ──
T1
4
T1
16
4
Meetkundige reeks. / Geometric series.
Meetkundige reeks. / Geometric series.
n − 1
n − 1
Tn = ar
Tn = ar
16 − 1
5
8 − 1
T16 = 2(2)
T8 = 16(──)
4
= 65 536
= 76,294
a(rn − 1)
a(rn − 1)
1.2
Sn = ───────
2.2
Sn = ───────
r − 1
r − 1
5
2(216 − 1)
16((──)8 − 1)
4
S16 = ───────
S8 = ───────
5
2 − 1
── − 1
4
= 131 070
= 317,470
1
T2
──
T2
3
1
64
3.1
a = T1 = 2 ; r = ── = ──
4.1
a = T1 = ─── r = ── = ── = 2
T1
2
1
128
T1
──
128
Tn = arn − 1
Tn = arn − 1
3
177 147
1
2(──)n − 1 = ──────
(───) X 2n − 1 = 512
2
2 048
128
3
3
(──) n − 1 = (──) 11
2−7 X 2n − 1 = 29
2
2
n − 1 = 11
n − 8 = 9
n = 12
n = 17
Term 12 is die laaste term. /
Term 17 is die laaste term. /
Term 17 is the last term.
Term 17 is the last term.
3
1
a(rn − 1)
2((──)12 − 1)
a(rn − 1)
(───)(217 − 1)
2
128
3.2
Sn = ────── :
Sn = ─────────
4.2
Sn = ────── :
Sn = ─────────
r − 1
2 − 1
r − 1
2 − 1
= 514,985
= 1 023,992
5.
Tn = arn − 1
6.
Tn = arn − 1
1
1
T4 = 54 : ar3 = 54 . . . (1)
T3 = ── : ar2 = ── . . . (1)
3
3
T6 = 486 : ar5 = 486 . . . (2)
T6 = 9 : ar5 = 9 . . . (2)
(2) ÷ (1) : r2 = 9
(2) ÷ (1) : r3 = 27
r = 3
r = 3
1
In / Into (1) : a(3)3 = 54
In / Into (1) : a(3)2 = ──
3
1
a = 2
a = ──
27
a(rn − 1)
a(rn − 1)
Sn = ───────
Sn = ───────
r − 1
r − 1
1
2(39 − 1)
──(311 − 1)
27
Sn = ───────
Sn = ───────
3 − 1
3 − 1
88573
= 19 682
= ────── = 3 280,481
27
Bereken die eeste drie terme en dan a en r. / First calculate the first three terms, then a and r.
3
7.
Tn = 5(2)k − 1
8.
Tn = (──)k + 1
2
3
9
T1 = 5(2)1 − 1 = 5
T1 = (──)1 + 1 = ──
2
4
3
27
T2 = 5(2)2 − 1 = 10
T2 = (──)2 + 1 = ───
2
8
3
81
T3 = 5(2)3 − 1 = 20
T3 = (──)3 + 1 = ───
2
16
20
10
9
81
27
3
a = 5 en / and r = ── = ── = 2
a = ── en / and r = ── ÷ ── = ──
10
5
4
16
8
2
a(rn − 1)
a(rn − 1)
Sn = ───────
Sn = ───────
r − 1
r − 1
9
3
5(26 − 1)
──((──)8 − 1)
4
2
S6 = ───────
S8 = ───────────
3
2 − 1
── − 1
2
18 915
= 315
= ────── = 73,887
256
Bereken die eerste drie terme en dan a en r.
First calculate the first three terms, then a and r.
As die onderste waarde nie 1 is nie, bereken
If the bottom value is not 1, then calculate
aantal terme so : Aantal
the number of terms like this : Number
terme = boonste waarde - onderste waarde + 1.
of terms = top value - bottom value + 1.
9.
Tn = 3.2n − 3
10.
Tn = 4.(−3) n − 2
T1 = 3.24 − 3 = 3.2 = 6
T1 = 4.(−3) 3 − 2 = 4.(−3) = −12
T2 = 3.25 − 3 = 3.22 = 12
T2 = 4.(−3) 4 − 2 = 4.(−3)2 = 36
T3 = 3.26 − 3 = 3.23 = 24
T3 = 4.(−3) 5 − 2 = 4.(−3)3 = −108
24
12
−108
36
a = 6 en / and r = ── = ── = 2
a = −12 en / and r = ──── = ──── = −3
12
6
36
−12
Aantal terme = 11 − 4 + 1 = 8
Aantal terme = 12 − 3 + 1 = 10
Number of terms = 11 − 4 + 1 = 8
Number of terms = 12 − 3 + 1 = 10
a(rn − 1)
a(rn − 1)
Sn = ───────
Sn = ───────
r − 1
r − 1
11
12
∑ 3.2n − 3 = S8
∑ 4.(−3) n − 2 = S10
n=4
n=3
6 (28 − 1)
−12 ((−3)10 − 1)
= ───────
= ────────────
2 − 1
−3 − 1
= 1 530
= 177 144
54
1
2.35
1
11.1
a = 55; r = ── = ─
12.1
a = 2.36; r = ──── = ─
55
5
2.36
3
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 and series converges.
∴ −1 < r < 1 and series converges.
a
55
a
2.36
11.2
S∞ = ──── = ─────
12.2
S∞ = ──── = ─────
1
1
1 − r
1 − ─
1 − r
1 − ─
5
3
15 625
= ───── = 3 906,25
= 2 187
4
/ 2 \
13.
Tn = | ─ | n + 1
14.
Tn = 43 − n
\ 3 /
/ 2 \
/ 2 \
4
T1 = | ─ | 1 + 1 = | ─ | 2 = ──
T1 = 43 − 1 = 42 = 16
\ 3 /
\ 3 /
9
/ 2 \
/ 2 \
8
T2 = | ─ | 2 + 1 = | ─ | 3 = ──
T2 = 43 − 2 = 41 = 4
\ 3 /
\ 3 /
27
/ 2 \
/ 2 \
16
T3 = | ─ | 3 + 1 = | ─ | 4 = ──
T3 = 43 − 3 = 40 = 1
\ 3 /
\ 3 /
81
4
2
1
a = ── ; r = ─
a = 16 ; r = ─
9
3
4
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 and series converges.
∴ −1 < r < 1 and series converges.
a(1 − rn)
a(1 − rn)
Sn = ────────
Sn = ────────
1 − r
1 − r
4
2
──(1 − (──) 8)
9
3
25 220
16
64
S8 = ───────────── = ───────
S10 = ──────── = ───
2
19 683
1
3
1 − ──
1 − ──
3
4
= 1,281
= 21,333
5
1
15.
Tn = 2.56 − n
16.
Tn = (5(──) − 4) 6 − n = (──) n
6
6
1
1
T1 = 2.56 − 1 = 2.55 = 6 250
T1 = (──) 1 = ──
6
6
1
1
T2 = 2.56 − 2 = 2.54 = 1 250
T2 = (──) 2 = ──
6
36
6
1
T3 = 2.56 − 3 = 2.53 = 250
T2 = (──) 3 = ───
6
216
1 250
1
1
1
a = 6 250 r = ──── = ─
a = ── r = ──
6 250
5
6
6
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 en reeks konvergeer.
∴ −1 < r < 1 and series converges.
∴ −1 < r < 1 and series converges.
a
a
1
6 250
──
6
S∞ = ───── = ─────
S∞ = ───── = ─────
1 − r
1 − r
1
1
1 − ──
1 − ──
5
6
1
= 7 812,5
= ──
5