NIMCET 2013 — Mathematics PYQ

Solution

The given series is:

Step 1: Group the terms in pairs

Since there are 40 terms (an even number), we can group them into 20 pairs:

Step 2: Apply the algebraic identity

Using the identity $a^2-b^2=(a-b)(a+b)$, we can simplify each pair. Notice that for every pair, $(a-b)$ will always be 1 (e.g., $2-1=1$, $4-3=1$).

• First pair: $(2-1)(2+1)=1(3)=3$

• Second pair: $(4-3)(4+3)=1(7)=7$

• Third pair: $(6-5)(6+5)=1(11)=11$

• Last pair: $(40-39)(40+39)=1(79)=79$

The series now becomes:

Step 3: Identify the Arithmetic Progression (AP)

The simplified series is an AP where:

• First term ($a$) = $3$

• Common difference ($d$) = $7-3=4$

• Number of terms ($n$) = $20$ (since we grouped 40 terms into pairs)

• Last term ($l$) = $79$

Step 4: Calculate the sum

Using the sum formula for an AP:

Final Answer: The sum is 820.