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Beskou die volgende meetkundige ry: 1 024 ; 256 ; 64 ; .. - NSC Mathematics - Question 3 - 2022 - Paper 1
Question 3
Beskou die volgende meetkundige ry: 1 024 ; 256 ; 64 ; ... Bereken: 3.1.1 Die 10de term van die ry 3.1.2 \[ \sum_{p=0}^{9} 256(4^{-p}) \] 3.2 Die eerste twee ter... show full transcript
Worked Solution & Example Answer:Beskou die volgende meetkundige ry: 1 024 ; 256 ; 64 ; .. - NSC Mathematics - Question 3 - 2022 - Paper 1
Step 1
Die 10de term van die ry
Answer
In a geometric sequence, the n-th term can be calculated using the formula:
[ T_n = ar^{(n-1)} ]
Here, we identify:
- First term, ( a = 1024 )
- Common ratio, ( r = \frac{256}{1024} = \frac{1}{4} )
- For the 10th term, ( n = 10 )
Now substituting into the formula:
[ T_{10} = 1024 \left( \frac{1}{4} \right)^{(10-1)} = 1024 \left( \frac{1}{4} \right)^{9} = 256 ]
Step 2
Die som van die eerste 10 terme
Answer
Using the formula for the sum of the first n terms of a geometric series:
[ S_n = \frac{a(1 - r^n)}{1 - r} ]
We substitute:
- First term, ( a = 1024 )
- Common ratio, ( r = \frac{1}{4} )
- Number of terms, ( n = 10 )
Calculating:
[ S_{10} = \frac{1024(1 - (\frac{1}{4})^{10})}{1 - \frac{1}{4}} = \frac{1024(1 - \frac{1}{1048576})}{\frac{3}{4}} = \frac{1024 \cdot \frac{1048575}{1048576}}{\frac{3}{4}} = 87381.33 ]
Step 3
Bepaal die waardes van t waarvoor die ry sal konvergeer
Answer
To find the values of t for which the geometric series converges, we first determine the ratio:
The general term is given as:
[ r = \frac{t^2 + 9t + 27 + 27}{2} ]
For convergence, the absolute value of the common ratio must be less than 1:
[ |r| < 1 ]
From our earlier simplification: [ -1 < \frac{t^2 + 9t + 27}{2} < 1 ]
Solving these inequalities gives the values of t. The simplified equation leads us to:
[ -5 < t < -1 ]