Question

In an Arithmetic Progression the 8th term plus the 4th term equals the 13th term. If the 18th term is 76 what is the sum of the first 20 terms.?

Answer 1

Step-by-step explanation:

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all in the sheet

Answer 2

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\large{\underline{\red{\rm{Let:}}}}$

In AP ,The first term=a

The common difference=d

$\large{\underline{\orange{\rm{Formula\: used:}}}}$

• $\small{\boxed{\red{\rm{T_n=a+(n-1)d}}}}$

• $\small{\boxed{\green{\rm{S_n=\frac{n}{2}[{2a+(n-1)d]}}}}}$

$\large{\underline{\purple{\rm{Given:}}}}$

• In an Arithmetic Progression the 8th term plus the 4th term equals the 13th term. If the 18th term is 76

$\small{\underline{\purple{\rm{As\:per\:the\:above\: information:}}}}$

$\sf{\longrightarrow 8th\:term+4th\: term=13th\: term}$

$\sf{\longrightarrow a+7d+a+3d=a+12d}$

$\sf{\longrightarrow 2a+10d=a+12d}$

$\sf{\longrightarrow 2a-a+10d-12d=0}$

$\sf\pink{\longrightarrow a-2d=0-----(1)}$

$\sf{\longrightarrow 18th\:term=76}$

$\sf{\longrightarrow a+17d=76}$

$\sf\pink{\longrightarrow a+17d=76-----(2)}$

• Now, Solving equation 1 and 2

$\sf{\longrightarrow a-2d=0}$

$\sf{\longrightarrow a+17d=76}$

• After solving we get,here

$\sf{\longrightarrow -19d=-76}$

$\sf{\longrightarrow 19d=76}$

$\sf\green{\longrightarrow d=4}$

• putting the value of d=4 in EQ 1

$\sf{\longrightarrow a-2d=0}$

$\sf{\longrightarrow a-2*4=0}$

$\sf\green{\longrightarrow a=8}$

• Now,The sum of 20th term of AP

$\sf{\longrightarrow S_2_0=\frac{20}{2}[2*8+(20-1)4]}$

$\sf{\longrightarrow S_2_0=\frac{20}{2}[16+76]}$

$\sf{\longrightarrow S_2_0=\frac{20}{2}*92}$

$\sf{\longrightarrow S_2_0=10*92}$

$\sf\green{\longrightarrow S_2_0=920}$

Hence,

$\small{\underline{\purple{\rm{The\:sum\:of\:1st\:20th\: term=920}}}}$