Math, asked by prakashinielambilass, 6 months ago

28. 23' term of an arithmetic sequence is 32 and its 35'h term is 104.
a) What is its common difference?
b) What is its first term?
c) Is the difference of any two terms of this sequence be 90?
d) Find the sum of the first 35 terms?​

Answers

Answered by brainlyofficial11
312

  { \huge✯ \: { \underline{ \underline  \red{ᴀɴsᴡᴇʀ }}} \: ✯ }

given

a_{23} = 32  \\ \:  \:  a_{35} = 104

solution

a) what is its common difference?

we know that,

 \boxed{\pink{\bold{a_{n} = a + (n - 1)d}}}

then,

 \bold{a_{23} = 32 }\: \:     { \tt\{here \: n = 23 \}} \:  \:  \:  \: \\  \\  \implies \: 32 = a + (23- 1)d  \:  \:  \:  \: \\  \\  \implies \: 32 = a  + 22d \: .......(i)

and,

 \bold{ a_{35} = 104} \:  \: { \tt \{here \: n = 35 \}} \:  \:  \:  \\  \\  \implies \: 104 = a + (35 - 1)d \:  \:  \:  \:  \:  \:  \\  \\  \implies \: 104 = a + 34d \: ........(ii)

now we have two equations,

32 = a + 22d ......(i)

104 = a + 34d ......(ii)

now, subtract eq.(i) from eq.(ii)

 { \tt➪ \:  \: a + 34d - (a + 22d) = 104 - 32} \\  \\ { \tt➪  \: \: a + 34d - a + 22d = 72} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\ { \tt➪ \:  \:  \cancel{a} + 34d -  \cancel{a} + 22d = 72} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\  { \tt➪ \:  \: 34d - 22d = 72} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\ { \tt➪ \:  \: 12d = 72} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   \\  \\  { \tt➪ \:  \: d =  \frac{72}{12} } \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\ ➪ \:  \:  {  \boxed {\orange{ \bold{d =6} }}} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

so,common difference of the AP is 6

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b) what is its first term?

from first part we get.

common difference(d) = 6

now, put the value of d in eq.(i)

 \bold{32 = a + 22d} \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\ { \tt➪ \:  \:  32 = a + 22 \times 6} \\  \\ { \tt➪ \:  \: 32 = a + 132} \:  \:  \:  \:  \:  \\  \\ { \tt➪ \:  \: a = 32 - 132} \:  \:  \:  \:  \\  \\ ➪ \:  \boxed{  \orange{ \bold {\: a =  - 100}}}\:  \:  \:  \:  \:  \:  \:  \:

so, first term of AP is -100

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c) is the difference of any two terms of this sequence be 90?

we know that,

common difference of this AP is 6

  { \tt and \:  \:  \: \:  \frac{90}{6} = 15 } \\

so, difference of nth term and (n+15)th term is 90

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for verification

☞ let n = 1 , then n+15 = 16

from 2nd part we have,

a_{1} =  - 100

now we have to find 16th of this AP

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \: a_{16} = a + (16 - 1)d \\  \\  \implies \: a_{16} =  - 100 + 15 \times 6 \\  \\  \implies \: a_{16} =  - 100 + 90 \:  \:  \:  \:  \:  \:  \:  \\  \\  \implies \: a_{16} =  - 10 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

now, see difference between these two terms

 \:  \:  \:  \:  \:  \:  \:  \:  \: a_{1} =  - 100 \\ \:  \:  \:  \:  \:  \:  \:  a _{16} =  - 10 \\  \\  { \tt \: and }\: \:  \:  \:  a_{16} - a_{1} \\  \\  \implies \:  - 10 - ( - 100) \\  \\  \implies \:  - 10 + 100 \:  \:  \:  \:  \:  \:  \:  \\  \\  \implies \:  90 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

hence, varified!

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so, we can say yes, here difference of nth term and (n+15)th term of this AP is 90

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d) find the sum of the first 35 terms?

we know that,

  \boxed{  \pink{\bold{ s_{n} = \frac{n}{2}   \{2a + (n - 1)d \}}}}

and here,

・a = -100

・d = 6

・n = 35

➪ \:  \: s_{35} =  \frac{35}{2}  \{ 2 \times  - 100 + (35 - 1)d\} \\  \\  ➪ \:  \: s_{35}  =  \frac{35}{2} \{ - 200 + 34 \times 6 \} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   \\  \\ ➪ \:  \: s_{35} =  \frac{35}{2}  \{  - 200 + 204\} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \\ ➪ \:  \: s_{35} =  \frac{35}{ \cancel{2}}  \times  \cancel{ 4}  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \\  \\  ➪ \:  \: s_{35} = 35 \times 2 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   \\  \\  ➪ \:  \:  \boxed{\orange{ \bold{ s_{35} = 70}}} \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

so, sum of first 35 terms of AP is 70

_________________________

hence,

  • a) common difference = 6
  • b) first term = -100
  • c) yes, difference of nth term and (n+15)th term of this AP is 90
  • d) sum of first 35 terms is 70
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