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131. Find the sum to n terms of the following series.

$space space space 1 cross times 2 cross times 3 plus 2 cross times 3 cross times 4 plus 3 cross times 4 cross times 5 plus........$

Here, $straight T subscript straight n equals end subscript$ [n^th term of 1, 2, 3,......] $cross times$ [n^th term of 2, 3, 4,.....] $cross times$ [n^th term of 3, 4, 5......]

= [1 + (n - 1). 1] [2 + (n - 1). 1] [3 + (n - 1) .1]

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Let $space space straight S subscript straight n$ denote the sum of n terms of series, then

$straight S subscript straight n space equals space sum from straight k equals 1 to straight n of straight T subscript straight k space equals space sum from straight k equals 1 to straight n of left parenthesis straight k cubed plus 3 straight k squared plus 2 straight k right parenthesis space equals space sum from straight k equals 1 to straight n of straight k cubed plus 3 space sum from straight k equals 1 to straight n of straight k squared plus 2 sum from straight k equals 1 to straight n of space straight k$

= $open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets squared plus 3. space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus 2. space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction$

$equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction plus fraction numerator 3. space left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 end fraction plus 2 close square brackets$

$equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis plus 2 left parenthesis 2 straight n plus 1 right parenthesis plus 2 cross times 2 over denominator 2 end fraction close square brackets$

$equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n squared plus straight n plus 4 straight n plus 2 plus 4 over denominator 2 end fraction close square brackets space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n squared plus 5 straight n plus 6 over denominator 2 end fraction close square brackets$

$space space equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis straight n squared plus 3 straight n plus 2 straight n plus 6 right parenthesis over denominator 4 end fraction$

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134 Views

132. Find the sum of n terms of the series 3 + 7 + 13 + 21 + 31 +.......

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133. The ages of the students in a class form an A.P. whose common difference is 4 months. If the youngest student is 8 years old and the sum of the ages of all the students in the class is 168 years. Find the numbers of students in the class.

787 Views

134.

A man deposited Rs. 10,000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

241 Views

135. An insect starts from a point and travels in a straight path 1 mm in the first second and half of the distance covered in the previous second in the succeeding second. In what time would it reach a point 3 mm away from its starting point?

234 Views

136. The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

270 Views

Long Answer Type

137. Shamshad Ali buys a scooter for Rs 22000. He pays Rs. 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

336 Views

Short Answer Type

138. A man buys every year Bank’s cash certificates of value exceeding the last year purchase by Rs. 25. After 20 years, he finds that the total value of the certificates purchased by him is Rs. 7250. Find the values of the certificates purchased by him

(i) In the first year. (ii) In the 13th year.

179 Views

Long Answer Type

139. An odd number of stones lie along a straight path, the distance between any two consecutive stones being 10 m. The stones are to the collected at the place where the middle stone lies. A man can carry only one stone at a time. He starts carrying the stones beginning from the extreme stone. If he covers a path of 3 km, how many stones are there ?

183 Views

Short Answer Type

140. The income of a person is Rs. 3,00,000 in first year and he receives an increase of Rs. 10,000 to his income per year for next 19 years. Find the total amount, he received in 20 years.

126 Views