A singer sang 100 songs on the first day, 90 songs on the second day and 80 songs on the third day, and so on in an arithmetic sequence. How many songs did the singer sing in a week? A. 490 B. 563 C. 575 D. 543
ANSWER
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Hint: As it is mentioned in the problem the number of songs sung by the singer follows the arithmetic progression. So find the different general terms associated with the A.P. like first term, common difference and number of terms from the statement and directly use the formula for sum of terms in A.P.
Complete step-by-step answer: According to the problem statement, the number of songs sung follows the A.P. So the A.P. is $100,90,80.....$ From the series we can see First term of the A.P. \[ = a = 100\] Common difference of the A.P. $ = d = \left( {{a_2} - {a_1}} \right) = \left( {90 - 100} \right) = - 10$ Number of terms = Number of days = 7 As we know the formula for sum of n number of A.P. is $ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ Where n= number of terms of A.P. a= first term of A.P. d=common difference of the A.P. So, putting in the values for the given question we get: \[ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right) \\ \Rightarrow {S_7} = \dfrac{7}{2}\left( {2 \times \left( {100} \right) + \left( {7 - 1} \right)\left( { - 10} \right)} \right) \\ \] Now, let us simplify the above term in order to find the sum \[ \Rightarrow {S_7} = \dfrac{7}{2}\left( {200 + 6\left( { - 10} \right)} \right) \\ \Rightarrow {S_7} = \dfrac{7}{2}\left( {200 - 60} \right) \\ \Rightarrow {S_7} = \dfrac{7}{2}\left( {140} \right) \\ \Rightarrow {S_7} = 7 \times 70 \\ \Rightarrow {S_7} = 490 \\ \] So, the sum of 7 numbers of A.P. is 490. Hence, the number of songs sung by the singer in a week is 490. So, option A is the correct option.
Note: An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second minus the first. For instance, the sequence 5, 7, 9, 11, 13, 15.. . is an arithmetic progression with a common difference of 2. Students must remember the formula for sum of ānā terms for different types of series.