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Formula for Sum of n Terms of an AP

Find the sum:...

Question

Find the sum:
(i) 2+4+6+...+200
(ii) 3+11+19+...+803
(iii) (-5)+(-8)+(-11)+...+(-230)
(iv) 1+3+5+7+...+199
(v) $7 + 10 \frac{1}{2} + 14 + . . . + 84$
(vi) 34+32+30+...+10
(vii) 25+28+31+...+100
(viii) $18 + 15 \frac{1}{2} + 13 + . . . + (- 49 \frac{1}{2})$

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Solution

$left parenthesis i i i right parenthesis space a space equals space minus 5 comma space d equals space minus 3a subscript n space equals space minus 230a subscript n space equals space a space plus space open parentheses n minus 1 close parentheses dminus 230 space equals negative 5 plus space open parentheses n minus 1 close parentheses left parenthesis negative 3 right parenthesiss o space n minus 1 space equals space fraction numerator negative 225 over denominator negative 3 end fraction space equals space 75n space equals 76S subscript n space equals space n over 2 open parentheses a space plus thin space a subscript n close parentheses space equals 76 over 2 open parentheses negative 5 plus left parenthesis negative 230 right parenthesis close parentheses spacerightwards double arrow S subscript n space equals space 38 cross times open parentheses negative 235 close parentheses space equals space minus 8930$

$left parenthesis v right parenthesis space a space equals space 7 comma space d equals space 3 1 half space equals space 3.5 a subscript n space equals space 84 a subscript n space equals space a space plus space open parentheses n minus 1 close parentheses d 84 space equals space 7 plus space open parentheses n minus 1 close parentheses 3.5 s o space n minus 1 space equals space fraction numerator 77 over denominator 3.5 end fraction space equals space fraction numerator 77 cross times 2 over denominator 3.5 cross times 2 end fraction space equals space 22 n space equals 23 S subscript n space equals space n over 2 open parentheses a space plus thin space a subscript n close parentheses space equals 23 over 2 open parentheses 7 plus 84 close parentheses space rightwards double arrow S subscript n space equals space fraction numerator 23 cross times 91 over denominator 2 end fraction space equals space 1046.5 space$

$left parenthesis v i right parenthesis space a space equals space 34 comma space d equals space minus 2 a subscript n space equals space 10 a subscript n space equals space a space plus space open parentheses n minus 1 close parentheses d 10 equals space 34 space plus space open parentheses n minus 1 close parentheses left parenthesis negative 2 right parenthesis s o space n minus 1 space equals space fraction numerator negative 24 over denominator negative 2 end fraction space equals space 12 n space equals 13 S subscript n space equals space n over 2 open parentheses a space plus thin space a subscript n close parentheses space equals 13 over 2 open parentheses 34 plus 10 close parentheses space rightwards double arrow S subscript n space equals space 13 cross times 22 space equals 286$

$left parenthesis v i i i right parenthesis space a space equals space 18 comma space d equals space minus 2 1 half space equals space minus 2.5 a subscript n space equals space minus 49.5 a subscript n space equals space a space plus space open parentheses n minus 1 close parentheses d minus 49.5 space equals space 18 space plus space open parentheses n minus 1 close parentheses left parenthesis negative 2.5 right parenthesis s o space n minus 1 space equals space fraction numerator negative 67.5 over denominator negative 2.5 end fraction space equals space 27 n space equals 28 S subscript n space equals space n over 2 open parentheses a space plus thin space a subscript n close parentheses space equals 28 over 2 open parentheses 18 minus 49.5 close parentheses space rightwards double arrow S subscript n space equals space 14 cross times left parenthesis negative 21.5 right parenthesis space equals 301$

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Similar questions

Find the sums given below

(i) 7 + + 14 + ………… + 84

(ii) 34 + 32 + 30 + ……….. + 10

(iii) − 5 + (− 8) + (− 11) + ………… + (− 230)

Re-arrange suitably and find the sum in each of the following :

(i) $\frac{11}{12} + \frac{- 17}{3} + \frac{11}{2} + \frac{- 25}{2}$

(ii) $\frac{- 6}{7} + \frac{- 5}{6} + \frac{- 4}{9} + \frac{- 15}{7}$

(iii) $\frac{3}{5} + \frac{7}{3} + \frac{9}{5} + \frac{- 13}{15} + \frac{- 7}{3}$

(iv) $\frac{4}{13} + \frac{- 5}{8} + \frac{- 8}{13} + \frac{9}{13}$

(v) $\frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5}$

(vi) $\frac{1}{8} + \frac{5}{12} + \frac{2}{7} + \frac{7}{12} + \frac{9}{7} + \frac{- 5}{16}$

Q. Simplify:
(i)

\frac{11}{12} \div (\frac{5}{9} \times \frac{18}{25})

(2 \frac{1}{2} \times \frac{8}{10}) \div (1 \frac{1}{2} + \frac{5}{8})

\frac{15}{16}

(\frac{5}{6} - \frac{1}{2}) \div \frac{10}{11}

\frac{9}{8} \div \frac{3}{5}

(\frac{3}{4} + \frac{3}{5})

\frac{2}{5} \div {\frac{1}{5} o f [\frac{3}{4} - \frac{1}{2}] - 1}

(1 \frac{3}{4} \times 3 \frac{1}{7}) - (4 \frac{3}{8} \div 5 \frac{3}{5})

Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data:

28	17	15	22	29	21	23	27	18	12	7	2	9	4
1	8	3	10	5	20	16	12	8	4	33	27	21	15
3	36	27	18	9	2	4	6	32	31	29	18	14	13
15	11	9	7	1	5	37	32	28	26	24	20	19	25
19	20	6	9

Q. Find the sum of each of the following arithmetic series:

(i)

7 + 10 \frac{1}{2} + 14 + . . . + 84

.
(ii) 34 + 32 + 30 + ... + 10.
(iii) (−5) + (−8) + (−11) + ... + (−230)
(iv) 5 + (–41) + 9 + (–39) + 13 + (–37) + 17 + ... + (–5) + 81 + (–3)

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