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3.1 Bewys dat $ ootnotesize{ extstyle{igg(m{ rac{ extstyle{ extstyle{igg(m{ rac{4}{3}}} + extstyle{ rac{4}{3}} + extstyle{ rac{4}{3}} + .. - NSC Mathematics - Question 3 - 2020 - Paper 1
Question 3
3.1 Bewys dat $ ootnotesize{ extstyle{igg(m{ rac{ extstyle{ extstyle{igg(m{ rac{4}{3}}} + extstyle{ rac{4}{3}} + extstyle{ rac{4}{3}} + ... + extstyle{ rac{4}... show full transcript
Worked Solution & Example Answer:3.1 Bewys dat $ ootnotesize{ extstyle{igg(m{ rac{ extstyle{ extstyle{igg(m{ rac{4}{3}}} + extstyle{ rac{4}{3}} + extstyle{ rac{4}{3}} + .. - NSC Mathematics - Question 3 - 2020 - Paper 1
Step 1
Bewys dat $\sum_{k=1}^{\infty} \frac{4}{3^k}$ 'n konvergerende meetkundige reeks is.
Answer
To prove that the series converges, we will use the formula for the sum of an infinite geometric series, which is given by:
where:
- (a) is the first term of the series,
- (r) is the common ratio of the series.
From the series ootnotesize{\sum_{k=1}^{\infty} \frac{4}{3^k}}:
- The first term, (a = \frac{4}{3} ), when ( k = 1).
- The common ratio, ( r = \frac{4/3^k}{4/3^{k-1}} = \frac{1}{3} ) for ( k \geq 1 ).
Since the magnitude of the common ratio must be less than 1 for convergence:
Thus, the series converges.
Step 2
Indien $\sum_{k=p}^{\infty} \frac{4}{3^k} = \frac{2}{9}, bepaal die waarde van p.$
Answer
We start with the general term for the sum of the series from ( k = p ):
where ( a = \frac{4}{3^p} ) and ( r = \frac{1}{3}.)
Thus, the sum becomes:
Setting this equal to ( \frac{2}{9} ):
By simplifying both sides, we find: