Math, asked by terabaap1292004, 8 months ago

If the 19th term of an AP is 39, find the sum of its first 37 terms​

Answers

Answered by BrainlyPopularman
8

GIVEN :

19th term of an AP = 39

TO FIND :

Sum of first 37 terms = ?

SOLUTION :

• If first term of A.P. is 'a' and common difference is d , then nth term –

  \\ \longrightarrow  \:  \:  \large { \boxed{ \sf  T_{n} = a + (n - 1)d}} \\

• According to the first condition –

  \\ \implies \sf  T_{19} = a + (19 - 1)d \\

  \\ \implies \sf  39 = a + 18d \\

  \\ \implies \sf   a + 18d = 39 \:  \:  -  -  - eq.(1) \\

• We know that Sum of n terms –

  \\ \longrightarrow  \:  \:  \large { \boxed{ \sf  S_{n} =  \dfrac{n}{2}[2a + (n - 1)d]}}\\

• So that –

  \\ \implies  \sf  S_{37} =  \dfrac{37}{2}[2a + (37 - 1)d] \\

  \\ \implies  \sf  S_{37} =  \dfrac{37}{2}[2a + 36d] \\

  \\ \implies  \sf  S_{37} =  \dfrac{37}{ \cancel2}[ \cancel2(a + 18d)] \\

  \\ \implies  \sf  S_{37} =  37(a + 18d) \\

• Using eq.(1) –

  \\ \implies  \sf  S_{37} =  37(39) \\

  \\ \implies \large { \boxed{ \sf  S_{37} =  1,443}} \\

Answered by Anonymous
50

{\huge{\bf{\red{\underline{Solution:}}}}}

{\bf{\blue{\underline{Given:}}}}

  • 19th term of an AP is 39.

{\bf{\blue{\underline{To\:Find:}}}}

  • Sum of its first 37th terms

{\bf{\blue{\underline{Formula \:Used:}}}}

  \dagger \:  \boxed{\sf{  a_{n}  = a + (n - 1)d}} \\ \\

  \dagger \:  \boxed{\sf{  S_{n}  =  \frac{n}{2}  2a+ (n - 1)d}} \\ \\

{\bf{\blue{\underline{Now:}}}}

 : \implies{\sf{  a_{19} = a + (19 - 1)d }} \\ \\

 : \implies{\sf{ 39 = a + (19 - 1)d }} \\ \\

 : \implies{\sf{ 39 = a + 18d }} \\ \\

__________________________________

 : \implies{\sf{  s_{37} =  \frac{37}{2}[ 2a + (37 - 1)d] }} \\ \\

 : \implies{\sf{  s_{37} =  \frac{37}{2} [2a + 36d ]}} \\ \\

 : \implies{\sf{  s_{37} =  \frac{37}{2} 2(a + 18d }} \\ \\

 : \implies{\sf{  s_{37} =  37( a + 18d )}} \\ \\

 : \implies{\sf{  s_{37} =  37( 39 )}} \\ \\

 : \implies{\sf{  s_{37} =  1443}} \\ \\

Similar questions