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In the sequence a1, a2, a3,…, an, …, each term after the fir
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06 Aug 2020, 05:29
06 Aug 2020, 05:29
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In the sequence \(a_1, a_2, a_3,…, a_n,\) …, each term after the first is equal to r times the previous term, where \(a_1 > 0\) and \(r > 1\). If \(a_1 + a_2 + a_3 = 21\) and \(a_2 + a_3 = 18\), what is the value of \(a_3 + a_4 + a_5\) ?
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84
Re: In the sequence a1, a2, a3,…, an, …, each term after the fir
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07 Aug 2020, 04:10
07 Aug 2020, 04:10
3
A = a1 + a2 + a3 = 21
B = a2 + a3 = 18
a1 = A - B = 3
Since every term is r times the previous term
a2 = a1 * r
a3 = a2 * r = a1 * r^2
B = a1 ( r + r^2 ) = 18
r (r + 1) = 18/3 = 6
so r = 2.
Now, a3 + a4 + a5 = a1 ( r^2 + r^3 + r^4 ) = 3(4 + 8 + 16) = 84.
Alternatively,
Once you realise r = 2 you can use the GP formula,
a3 + a4 + a5 = S5 (sum of first 5 terms) - S2 (sum of first 2 terms) = 93 - 9 = 84
B = a2 + a3 = 18
a1 = A - B = 3
Since every term is r times the previous term
a2 = a1 * r
a3 = a2 * r = a1 * r^2
B = a1 ( r + r^2 ) = 18
r (r + 1) = 18/3 = 6
so r = 2.
Now, a3 + a4 + a5 = a1 ( r^2 + r^3 + r^4 ) = 3(4 + 8 + 16) = 84.
Alternatively,
Once you realise r = 2 you can use the GP formula,
a3 + a4 + a5 = S5 (sum of first 5 terms) - S2 (sum of first 2 terms) = 93 - 9 = 84
In the sequence a1, a2, a3,, an, , each term after the fir
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12 Aug 2022, 18:01
12 Aug 2022, 18:01
1
Same solution... some more details
a1 + a2 + a3 = 21 ----> (1)
a2 + a3 = 18 -----> (2)
From Eqn (1) & (2),
a1 = 21-18 = 3
Now, since each term after the first equals r times the previous term, we have:
a2 = a1 * r
a3 = a2 * r = a1 * r^2
Plugging the values into a2 & a3 in Eqn (2)
a1*r + a1*r^2 = 18
3 * (r + r^2) = 18
r(r+1) = 6
r^2 + r - 6 = 0
(r - 2)(r + 3) = 0
r = -3 or 2
So we know it must be 2.
Then we can solve the question:
a3 + a4 + a5 = a1 * (r^2 + r^3 + r^4)
= 3 * (4 + 8 + 16)
= 3 * 28
= 84
a1 + a2 + a3 = 21 ----> (1)
a2 + a3 = 18 -----> (2)
From Eqn (1) & (2),
a1 = 21-18 = 3
Now, since each term after the first equals r times the previous term, we have:
a2 = a1 * r
a3 = a2 * r = a1 * r^2
Plugging the values into a2 & a3 in Eqn (2)
a1*r + a1*r^2 = 18
3 * (r + r^2) = 18
r(r+1) = 6
r^2 + r - 6 = 0
(r - 2)(r + 3) = 0
r = -3 or 2
So we know it must be 2.
Then we can solve the question:
a3 + a4 + a5 = a1 * (r^2 + r^3 + r^4)
= 3 * (4 + 8 + 16)
= 3 * 28
= 84
