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