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1 year ago

What is the 20th term of 64,67,70,73,76

Mathematics

1 answer:

PIT_PIT [208]1 year ago

6 0

Let's analyze the increments for each step of the sequence:

Each step we add 3 to the previous number.

Since we need the 20th, from what we saw, in the 20th term we will have added three 20 times.

So the procedure is: calculate how much is 3 times 20 and then add that to the first term of the sequence.

$3\times20=60$

So we add 60 to the first term to find the 20th term:

$64+60=124$

Answer: 124

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and subtracting this from $S_n$ gives

$(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}$

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$a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a$

$a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}$

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Solving for r by substitution, we have

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Recalling the difference of squares identity, we have

$\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}$

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$\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15$

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