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  • Which term of the AP: -2, -7, -12,... will be -77? Find the sum of this AP upto the term -77.

Which term of the AP: -2, -7, -12,... will be -77? Find the sum of this AP upto the term -77.

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\text{The given AP is } -2, -7, -12, \ldots \\ \text{Here first term }(a) = -2 \\ \text{Common difference }(d) = -7 - (-2) = -7 + 2 = -5 \\ \text{Let } n\textsuperscript{th} \text{ term be } -77 \\ \text{Then } a_{n} = -77 \\ a + (n - 1)d = -77 \\ -2 + (n - 1)(-5) = -77 \\ (n - 1)(-5) = -77 + 2 \\ (n - 1) = \frac{-75}{-5} = 15 \\ n = 15 + 1 \\ n = 16 \\ \text{Hence, the } 16\textsuperscript{th} \text{ term is } -77 \\ \text{Sum of 16 terms}

\begin{aligned} \mathrm{S}_{16} &=\frac{16}{2}[2(-2)+(16-1)(-5)] \quad\left(\text { Using } \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]\right) \\ &=8[-4+15(-5)] \\ &=8[-4-75] \\ \mathrm{S}_{16} &=8 \times-79 \\ \mathrm{~S}_{16}=&-632 \end{aligned}

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