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'n Konvergente meetkundige reeks wat slegs uit positiewe terme bestaan, het eerste term a, konstante verhouding r en n ext{de} term, T_n, soadat $$ extstyleegin{aligned} extstyle extstyle extstyle extstyle extstyle S_n = rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle S_n extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle ight) = rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle $$ 3.1 Indien $T_1 + T_2 = 2$, skryf 'n uitdrukking vir $a$ in terme van $r$ neer - NSC Mathematics - Question 3 - 2017 - Paper 1
Question 3
'n Konvergente meetkundige reeks wat slegs uit positiewe terme bestaan, het eerste term a, konstante verhouding r en n ext{de} term, T_n, soadat $$ extstyl... show full transcript
Worked Solution & Example Answer:'n Konvergente meetkundige reeks wat slegs uit positiewe terme bestaan, het eerste term a, konstante verhouding r en n ext{de} term, T_n, soadat $$ extstyleegin{aligned} extstyle extstyle extstyle extstyle extstyle S_n = rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle S_n extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle ight) = rac{1}{4} extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle $$ 3.1 Indien $T_1 + T_2 = 2$, skryf 'n uitdrukking vir $a$ in terme van $r$ neer - NSC Mathematics - Question 3 - 2017 - Paper 1
Step 1
Indien $T_1 + T_2 = 2$, skryf 'n uitdrukking vir $a$ in terme van $r$ neer.
Answer
In a geometric series, the first term is denoted as ( a ) and the common ratio as ( r ). The first two terms are given by:
( T_1 = a ) and ( T_2 = ar ).
Given that ( T_1 + T_2 = 2 ), we can substitute:
[ a + ar = 2 ]
Factoring out ( a ):
[ a(1 + r) = 2 ]
Thus,
[ a = \frac{2}{1 + r} ]
Step 2
Bereken die waardes van $a$ en $r$.
Answer
Using the formula for the sum of the first n terms of a geometric series:
[ S_n = T_1 + T_2 + \sum_{n=3}^{n} T_n ]
We have:
[ S_n = a + ar + \frac{ar^2}{1 - r} ]
Given ( S_n = \frac{1}{4} ), substituting ( a = \frac{2}{1 + r} ):
[ \frac{2}{1 + r} + \frac{2r}{1 + r} + \frac{\frac{2r^2}{1 + r}}{1 - r} = \frac{1}{4} ]
Simplifying gives:
[ 2 + 2r + \frac{2r^2}{(1 - r)(1 + r)} = \frac{1}{4} ]
This leads to a quadratic equation in terms of ( r ):
Substituting values gives possible values of ( r = \frac{1}{3} ) and consequently ( a = \frac{3}{2} ).