Art, asked by qoz, 2 months ago

Find the sum of the positive terms of the arithmetic sequence 85, 78, 71, ...​

Answers

Answered by XxItsPriNcexX
3

\huge✍︎\fbox\orange{ÂŇ}\fbox{SW} \fbox\green{ÊŘ}:

Given: \\ </p><p> \\ </p><p>A.P = 85, 78, 71, . . . . \\ </p><p> \\ </p><p>The  \: first \:  term \:  of \:  the  \: A.P (a_1)  \: is \:  85 \\ </p><p> \\ </p><p>Now, \\ </p><p> \\ </p><p>78 - 85 = -7 \\ </p><p> \\ </p><p>71 - 78 = -7</p><p> \\ </p><p>Common \:  difference \:  is  \: (d) = -7 \\ </p><p> \\ </p><p>By \:  looking  \: at \:  the  \: further  \: numbers \\  of  \: the \:  A.P,  \: we \:  get \:  to \:  know \:  that \:  (7) \\  decreasing  \: in  \: each \:  number, \:  after  \\ some \:  point  \: negative \:  number \:  will \\  arise, \\ </p><p> \\ </p><p>We \:  know \:  that, \\ </p><p> \\ </p><p>a_n \: = a + (n-1)d \\ </p><p> \\ </p><p>a_{10} = 85 + (10 - 1)(-7) \\ </p><p> \\ </p><p>a_{10}= 85 + (9)(-7) \\ </p><p> \\ </p><p>a_{10} = 85 - 63 \\ </p><p> \\</p><p>a_{10} = 22 \\ </p><p> \\ </p><p>And, \\ </p><p> \\ </p><p>a_{15} = 85 + (15 - 1)(-7) \\ </p><p> \\ </p><p>a_{15} = 85 + (14)(-7) \\ </p><p> \\ </p><p>a_{15} = 85 - 98 \\ </p><p> \\ </p><p>a_{15}= -13 \\ </p><p> \\ </p><p>And, \\ </p><p> \\ </p><p>a_{13} = 85 + (13 - 1)(-7) \\ </p><p> \\ </p><p>a_{13} = 85 + (12)(-7) \\ </p><p> \\ </p><p>a_{13} = 85 - 84 \\ </p><p> \\ </p><p>a_{13} = 1 \\ </p><p> \\ </p><p>And, \\ </p><p> \\ </p><p>a_{14} = 85 + (14 - 1)(-7) \\ </p><p> \\ </p><p>a_{14} = 85 + (13)(-7) \\ </p><p> \\ </p><p>a_{14} = 85 - 91 \\ </p><p> \\ </p><p>a_{14} = -6 \\ </p><p> \\ </p><p>Therefore, \\ </p><p> \\ </p><p>a_{13} = 1  \: and \:  a_{14} = -6 \\ </p><p> \\ </p><p>So,  \: till  \: a_{13} = 1  \: positive \\  terms \:  are  \: there, \\ </p><p> \\ </p><p>We \:  have  \: to  \: find \:  the \:  sum  \: of  \: 13 \\  positive \:  numbers. \\ </p><p> \\ </p><p>Formula, \\ </p><p> \\ </p><p>\boxed{\sf S_{n}=\frac{n}{2}[2a+(n-1)d]} \\</p><p> \\ </p><p>\sf S_{13}=\frac{13}{2}[2(85)+(13-1)(-7)]  \\ </p><p> \\ </p><p>\sf S_{13}=\frac{13}{2}[170+(12)(-7)]  \\ </p><p> \\ </p><p>\sf S_{13}=\frac{13}{2}[170-84] \\ </p><p> \\ </p><p>\sf S_{13}=\frac{13}{2}\times 86 \\ </p><p> \\ </p><p>\sf S_{13}=13 \times 43 \\ </p><p> \\ </p><p>\sf S_{13}=559\\ </p><p> \\ </p><p>Hence, \\ </p><p> \\ </p><p>The \:  sum  \: of \:  the \:  positive \:  terms   \\ of  \: the \: \orange { A.P} \:  is \:   \green{559} \\ </p><p></p><p>

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