28. 23' term of an arithmetic sequence is 32 and its 35'h term is 104.

1.	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?
Answer» ${ \huge✯ \: { \underline{ \underline \red{ᴀɴsᴡᴇʀ }}} \: ✯ }$ ★ given → $a_{23} = 32 \\ \: \: a_{35} = 104$ ★ solution → a) what is its COMMON difference? we know that, $\boxed{\pink{\bold{a_{n} = a + (n - <klux>1</klux>)d}}}$ 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 _________________________ 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 _________________________ 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 _______________________________ _______________________________ ✩ 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! _______________________________ _______________________________ so, we can say yes, here difference of nth term and (n+15)th term of this AP is 90 _________________________ 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

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?

Answer»

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

★ given →

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

★ solution →

a) what is its COMMON difference?

we know that,

$\boxed{\pink{\bold{a_{n} = a + (n - <klux>1</klux>)d}}}$

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

_________________________

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

_________________________

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

_______________________________

✩ 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!

_______________________________

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

_________________________

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

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?

★ given →