Question

Sol. [Key Concept: Terms in A.P. ⇒ use the common-difference relations.]

(b) $\left( a_{1} + a_{5} \right) - 2\left( a_{2} + a_{4} \right) + 2a_{3} = 2a_{3} - 4a_{3} + 2a_{3} = 0$

$$\text{~}\text{(d)}\text{~}\begin{matrix} a_{1} - & 4a_{2} + 6a_{3} - 4a_{4} + a_{5} + d \\ & \ = \left( a_{1} + a_{5} \right) - 4\left( a_{2} + a_{4} \right) + d + 6a_{3} = 2a_{3} - 8a_{3} + d + 6a_{3} = 0 + d = d \end{matrix}$$

✔ (a) $a_{1 - 2}a_{2 +}a_{3 = 0}$ (2nd difference)

✔ (b) $a_{1 - 2}a_{2 + 2}a_{3 - 2}a_{4 +}a_{5 = 0}$

✘ (c) does not vanish

✘ (d) $= d \neq 0$