Question

In an AP :
Given a = 2, d = 8, Sn 90

find n and an​

Answer 1

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{GIVEN}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\:\:\:\sf\blue{a\:=\:2}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\:\:\:\sf\blue{d\:=\:8}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\:\:\:\sf\blue{S_{n}\:=\:90}$

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{TO\:FIND}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\:\:\:\sf\blue{n\:and\:a_{n}}$

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{SOLUTION}$

$\dagger\:\:\boxed{\bold{\pink{S_{n}\:=\:\frac{n}{2} [2a\:+\:(n\:-\:1)d]}}}$

$\large\leadsto\:\:\: \: \sf \purple {90={\Large\frac{n}{2}}[2×2+(n-1)×8]}$

$\large\leadsto\:\:\: \: \sf \green {90={\Large\frac{n}{2}}[4+8n-8]}$

$\large\leadsto\:\:\: \: \sf \purple {90={\Large\frac{n}{2}}[8n-4]}$

$\large\leadsto\:\:\: \: \sf \green {90={\Large\frac{4n}{2}}[2n-1]}$

$\large\leadsto\:\:\: \: \sf \purple {90=2n[2n-1]}$

$\large\leadsto\:\:\: \: \sf \green {4n^2 - 2n = 90}$

$\large\leadsto\:\:\: \: \sf \purple {4n^2 - 2n -90=0}$

$\large\leadsto\:\:\: \: \sf \green {2(2n^2 -n-45)=0}$

$\large\leadsto\:\:\: \: \sf \purple {2n^2 -n- 45 = 0}$

$\large\leadsto\:\:\: \: \sf \green {2n^2 -10n+9n-45=0}$

$\large\leadsto\:\:\: \: \sf \purple {2n(n-5)+9(n-5)=0}$

$\large\leadsto\:\:\: \: \sf \green {(n-5)(2n+9)=0}$

$\large\leadsto\:\:\: \: \sf \purple {n-5=0\:(or)\:2n+9=0}$

$\large\leadsto\:\:\: \: \sf \green {n=5 \:(or)\:n=\frac{-9}{2}}$

$\Large\leadsto\:\:\: \: \sf \underline\orange {n\:=\:5}$

$\Large\star\:\:\:\sf\red{a_n = a_5 = a + 4d}$

$\Large\leadsto\:\:\: \: \sf \purple {a_n = 2+4×8}$

$\Large\leadsto\:\:\: \: \sf \green {a_n = 2+32}$

$\Large\leadsto\:\:\: \: \sf \underline\orange {a_n \: = \: 34}$

$\rule{200}{2}$