{"id":18748,"date":"2018-01-18T04:41:40","date_gmt":"2018-01-18T04:41:40","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=18748"},"modified":"2020-11-26T11:50:13","modified_gmt":"2020-11-26T06:20:13","slug":"math-labs-activity-sum-first-n-terms-ap","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/math-labs-activity-sum-first-n-terms-ap\/","title":{"rendered":"Math Labs with Activity – Sum of the First n Terms of an AP"},"content":{"rendered":"
OBJECTIVE<\/strong><\/span><\/p>\n To verify that the sum of the first n terms of an arithmetic progression where a is the first term and d is the common difference is given by Materials Required<\/strong><\/span><\/p>\n Theory<\/strong> <\/span> Procedure<\/strong> <\/span> Observations and Calculations<\/strong><\/span><\/p>\n Result<\/strong> <\/span> Math Labs with Activity<\/a>Math Labs<\/a>Science Practical Skills<\/a>Science Labs<\/a><\/p>\n Math Labs with Activity<\/a>Math Labs<\/a>Science Practical Skills<\/a>Science Labs<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Math Labs with Activity – Sum of the First n Terms of an AP OBJECTIVE To verify that the sum of the first n terms of an arithmetic progression where a is the first term and d is the common difference is given by Materials Required A sheet of white paper A geometry box A … Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6805],"tags":[],"yoast_head":"\n
\n<\/p>\n\n
\nIf a is the first term, d the common difference and l the nth term of an AP then
\nl=a + (n-1)d. … (i)
\nNow, the sum of n terms of an AP is given by
\n
\n[using equation (i)].<\/p>\n
\nStep 1:<\/strong> We shall verify the above formula for a general AP having the first term a and the common difference d for n = 10.
\nStep 2:<\/strong> Draw horizontal lines on the sheet of paper with a distance of 1 unit between two consecutive lines.
\nStep 3:<\/strong> Cut 10 small rectangular strips from the coloured paper tape, each of the same length (say, a units).
\nStep 4:<\/strong> Cut 45 other small rectangular strips from the paper tape, each of the same length (say, d units).
\nStep 5:<\/strong> Paste both types of strips on the white paper along the horizontal lines so as to obtain rectangles of lengths a,a + d,a + 2d,…,a+9d arranged sequentially, as shown in Figure 3.1.
\nStep 6:<\/strong> Extend the line DE to C by a units to construct the rectangle ABCD (as shown in Figure 3.1).
\nStep 7:<\/strong> Cut the portion of the rectangle ABCD which is covered with the coloured paper tape. We find that this portion completely covers the remaining portion of the rectangle ABCD.
\n<\/p>\n\n
\n\u2234\u00a0 the area of the rectangle ABCD = 10(2a + 9d) units\u00b2 … (ii)<\/li>\n
\n= (a x 1) + [(a + d) x 1] + [(a + 2d) x 1]+…+ [(a + 9d) x 1]
\n= a+(a + d)+(a + 2d) +…+ (a + 9d). … (iii)<\/li>\n
\na + (a + d)+(a + 2d) +…+(a + 9d) = 10\/2 (2a + 9d) [using equations (ii) and (iii)]
\ni.e., a + (a + d) + (a + 2d) +…+ [a + (n -1 )d] = n\/2 [2a+(n -1 )d] for n = 10.<\/li>\n<\/ol>\n
\nIt is verified for n = 10 that the sum of the first n terms of an AP is given by
\n
\nRemarks:<\/strong>
\nThe students shall apply the above method of verification for various values of n, taking different values of a and d as well.<\/p>\n