15/20
>
33 min 15 secs
75%
For a negatively skewed
distribution, what is the
most probable order for
the three measures of
central tendency from
smallest to largest?
Mode, mean, median
Median, Mode, Mean
Mode, median, mean
Mean, median, mode
Mean, mode, median
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.