Most Important Selected Qs for JEE AdvancedMathematicsInverse Trigonometric Functions
Let S_n= _ k=1 ^n ⁻¹ ( 1 k(k+1)+1 ) for positive integers n N , then:
Options
- Athe value of S₁₀ is equal to 4 - ⁻¹ ( 1 11 ) .
- Bthe value of _ n S_n is equal to 2 .
- Cthe value of 5+ _ n=1 ⁶² 1+ S_n 1- S_n is equal to 2020
- Dthe value of S₅ is equal to ⁻¹(6)- 4 .
Correct answer
D. the value of S₅ is equal to ⁻¹(6)- 4 .
Step-by-step solution
aligned & S_n= _ k=1 ^n ⁻¹ 1 k(k+1)+1 & S_n= _ k=1 ^n ⁻¹ 1 k(k+1) 1+ 1 k(k+1) & S_n= _ k=1 ^n ⁻¹ 1 k - 1 k+1 1+ 1 k(k+1) & S_n= _ k=1 ^n ( ⁻¹ 1 k - ⁻¹ 1 k+1 ) aligned aligned S_n= ⁻¹ 1 1 - ⁻¹ 1 2 + ⁻¹ 1 2 - ⁻¹ 1 3 + .+ ⁻¹ & 1 n-1 - ⁻¹ 1 n & + ⁻¹ 1 n - ⁻¹ 1 n+1 aligned S_n= ⁻¹ 1- ⁻¹ 1 n+1 5+ _ n=1 ⁶² 1+ S_n 1- S_n =5+ _ n=1 ⁶² ( 4 +S_n )=5+ _ n=1 ⁶² ( 4 + 4 - ⁻¹ 1 n+1 ) aligned & =5+ _ n=1 ⁶² ( 2 - ⁻¹ 1 n+1 )=5+ _ n=1 ⁶² ( ⁻¹ 1 n+1 ) & =5+ _ n=1 ⁶² 1 ( ⁻¹ 1 n+1 ) =5+ _ n=1 ⁶² 1 1 n+1 =5+ _ n=1 ⁶²(n+1) & =4+(1+2+3+ .