Math, asked by ash963cash, 1 month ago

Q3. FIND X IF angle = angle 2 and angle 3- angle 4
30
A
1
3
4
O 15
O 40
60
020​

Answers

Answered by Shanu1982
0

Step-by-step explanation:

Answer:-

Let the first term be a and common ratio be r.

Given:-

Sum of first 20 terms of GP = 244 × Sum of first 10 terms.

We know that,

\sf \: Sum \: of \: first \: n \: terms \: (S_n) = \dfrac{a( {r}^{n} - 1) }{r - 1}Sumoffirstnterms(S

n

)=

r−1

a(r

n

−1)

So,

\begin{gathered} \implies \sf \: \dfrac{a( {r}^{20} - 1)}{r - 1} = 244 \times \dfrac{a( {r}^{10} - 1)}{r - 1} \\ \\ \\ \implies \sf \: \dfrac{a( {r}^{20} - 1)}{(r - 1)} \times \frac{(r - 1)}{a( {r}^{10} - 1)} = 244 \\ \\ \\ \implies \sf \: \frac{ {r}^{20} - 1}{ {r}^{10} - 1} = 244 \\ \\ \\\implies \sf \: \frac{ {( {r}^{10}) }^{2} - 1 }{ {r}^{10} - 1} = 244 \: \: \: \: \: ( \because \: {( {a}^{m} )}^{n} = {a}^{mn} )\end{gathered}

r−1

a(r

20

−1)

=244×

r−1

a(r

10

−1)

(r−1)

a(r

20

−1)

×

a(r

10

−1)

(r−1)

=244

r

10

−1

r

20

−1

=244

r

10

−1

(r

10

)

2

−1

=244(∵(a

m

)

n

=a

mn

)

Let r¹⁰ = a.

\begin{gathered} \: \implies \sf \: \frac{ {a}^{2} - 1}{a - 1} = 244 \\ \\ \\ \implies \sf \: {a}^{2} - 1 = 244(a - 1) \\ \\ \\ \implies \sf \: {a}^{2} - 1 = 244a - 244 \\ \\ \\ \implies \sf \: {a}^{2} - 244a - 1 + 244 = 0 \\ \\ \\ \implies \sf \: {a}^{2} - 244a + 243 = 0 \\ \\ \\ \implies \sf \: {a}^{2} - 243a - a + 243 = 0 \\ \\ \\ \implies \sf \:a(a - 243) - 1(a - 243) = 0 \\ \\ \\ \implies \sf \:(a - 1)(a - 243) = 0 \\ \\ \\ \implies \red{\sf \:a = 1 \: (or)\: 243}\end{gathered}

a−1

a

2

−1

=244

⟹a

2

−1=244(a−1)

⟹a

2

−1=244a−244

⟹a

2

−244a−1+244=0

⟹a

2

−244a+243=0

⟹a

2

−243a−a+243=0

⟹a(a−243)−1(a−243)=0

⟹(a−1)(a−243)=0

⟹a=1(or)243

Hence,

⟹ a = r¹⁰ = 1 , 243

⟹ r¹⁰ = (± 1)¹⁰ , (√3)¹⁰.

[ ∵ 3⁵ = 243 ⟹ (√3 × √3)⁵ = [(√3)² ]⁵ = (√3)¹⁰ ]

⟹ r = ± 1 , √3.

∴ Common ratio of the given GP is ± 1 or √3.

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