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  • For the AP: –3, –7, –11, ..., can we find directly a_30 – a_20 without actually finding a_30 and a_20? Give reasons for your answer.

For the AP: –3, –7, –11, ..., can we find directly a_{30} - a_{20} without actually finding a_{30} \text{ and } a_{20}?

Give reasons for your answer.

Answers (1)

Answer.  [True]
Solution. Here the given series is

-3,-7,-11 \ldots
 a_{1}=-3, a_{2}=-7
\begin{align*} \mathrm{d} &= \mathrm{a}_{2} - \mathrm{a}_{1} \\ &= -7 - 3 \\ &= -7 + 3 \\ &= -4 \\ \mathrm{a}_{30} - \mathrm{a}_{20} &= \mathrm{a} + (30 - 1) \mathrm{d} - (\mathrm{a} + (20 - 1) \mathrm{d}) \\ &= \mathrm{a} + 29 \mathrm{~d} - \mathrm{a} - 19 \mathrm{~d} \\ &= 10 \mathrm{~d} \\ &= 10 \times (-4) \\ &= -40 \end{align*}
\text{Yes we can find} \mathrm{a}_{30}-\mathrm{a}_{20}  \text{without actually calculating} \mathrm{a}_{30} \text{ and } \mathrm{a}_{20}

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