Sequence And Series

Concept: Nth term from sum formula: Step 1: {\color{red}Compute general term.} So: Step 2: {\color{red}Use given term.}

Concept: Sum of first terms of A.P.: Sum of last terms = difference of sums. Step 1: {\color{red}Use first four terms sum.} Given : Step 2: {\color{red}Let total terms = .} Sum of last four terms: Now: Substitute : Similarly: Step 3: {\color{red}Use last four terms sum.} Expand: Closest intended option ⇒ .

Concept: Let: Given: This condition restricts the polynomial. Step 1: {\color{red}Use given condition.} So polynomial becomes: Step 2: {\color{red}Find derivative.} This is a linear function in . Step 3: {\color{red}Use A.P. property.} If are in A.P., then: Now: Since multiplying an A.P. by constant preserves A.P., are also in A.P.

Concept: Since are in A.P.: Step 1: {\color{red}Substitute into equations.} First equation simplifies using A.P. relation. Common root condition implies discriminant consistency. Step 2: {\color{red}Use proportional coefficients.} Common root ⇒ equations share a factor. Equating ratios of coefficients gives: So squares also in A.P.

Concept: General term: Power of : For constant term: Step 1: {\color{red}Find constant term.} Step 2: {\color{red}Substitute into expression.} Step 3: {\color{red}Evaluate limit.} Using: Hence limit → .