| Class XI |
Mathematics |
Mathematics |
8 |
a , a 8. a = n ;a 7 n 17 24 n 2 n(n – 2) 9. a = (–1) n – n ; a 10. a = ;a . n 9 n n+3 20 Reprint 2026-27 SEQUENCES AND SERIES 139 Write the first f... |
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| Class XI |
Mathematics |
Mathematics |
8 |
a , a , a , …, a , … is called geometric progression, if each term is 1 2 3 n ak +1 non-zero and a = r (constant), for k ≥ 1. k 2 3 By letting a 1 ... |
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| Class XI |
Mathematics |
Mathematics |
8 |
suggests that the n term of a G.P. is given by n–1 a n ar . 2 3 n – 1 2 n – 1 Thus, a, G.P. can be written as a, ar, ar , ar , … ar ; a, ar, ar ,..... |
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| Class XI |
Mathematics |
Mathematics |
8 |
= 4. Therefore 131072 = an= 2(4)n – 1 or 65536 = 4n – 1 8 n – 1. Thisgives 4 = 4 So that n – 1 = 8, i.e., n = 9. Hence, 131072 is the 9 term of the... |
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| Class XI |
Mathematics |
Mathematics |
8 |
, which gives n = 10. Example 9 The sum of first three terms of a G.P.and their product is – 1. Find the common ratio and the terms. Solution Leta ... |
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| Class XI |
Mathematics |
Mathematics |
8 |
number of ancestors preceding the person is 2046. 8.4.3 Geometric Mean (G.M.) The geometric mean of two positive numbers a and b is the number . Th... |
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| Class XII |
Accountancy |
Computerised Accounting System |
2 |
Fill command can be used to fill data into worksheet cells (Figure 2.36 & 2.37. The Excel provides for entering data automatically to continue a se... |
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| Class XI |
Mathematics |
Mathematics |
8 |
G.M. Let AandGbeA.M.andG.M.oftwogivenpositiverealnumbersaandb,respectively. Then A= a+b andG = ab Thus, we have A – G = a+b − ab = a+b−2 ab 2 2 ( a... |
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| Class XI |
Mathematics |
Mathematics |
8 |
to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 7. 0.15, 0.015, 0.0015, ... 20 terms. 8. 7 , 21 , 3 7 , ..... |
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| Class XI |
Mathematics |
Mathematics |
8 |
AR, AR , … AR form a G.P, and find the common ratio. 21. Find four numbers forming a geometric progression in which the third term is greater than ... |
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| Class XI |
Mathematics |
Mathematics |
8 |
2026-27 SEQUENCES AND SERIES 147 31. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% com... |
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| Class XII |
Informatics Practices |
Informatics Practices |
1 |
value as sum of Price and 12% of the GST. Reprint 2026-27 6 NFORMATICSPRACTICES mysql> ALTER TABLE INVENTORY ADD FinalPrice Numeric(10,1); Query OK... |
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| Class XI |
Mathematics |
Mathematics |
8 |
terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms. 3. The first term... |
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| Class XI |
Mathematics |
Mathematics |
8 |
(i) 5 + 55 +555 + … (ii) .6 +. 66 +. 666+… 12. Find the 20 term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms. 13. A farmer buys a used tract... |
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| Class XI |
Mathematics |
Mathematics |
8 |
number of terms is called a finite sequence. Asequence is called infinite if it is not a finite sequence. fi Let a 1 a2, 3 , ... be the sequence, th... |
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| Class XI |
Mathematics |
Mathematics |
8 |
into specific forms. In 1671 James Gregory used the term infinite series in connectionwithinfinitesequence.Itwasonlythroughtherigorousdevelopmentof... |
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| Class XI |
Mathematics |
Mathematics |
9 |
Chapter 9 STRAIGHTLINES v G eometry, as a logical system, is a means and even the most powerful means to make children feel the strength of the hum... |
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| Class XI |
Mathematics |
Mathematics |
9 |
m y 2+ n y1 and (x 2 y2) internally, in the ratio m: n re , . m + n m + n For example, the coordinates of the point which divides the line ... |
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| Class XI |
Mathematics |
Mathematics |
9 |
to x-axis, or coincidingwithx-axis,haveinclinationof0°.The inclination of a vertical line (parallel to or coincidingwith y-axis)is90°. Definition 1... |
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| Class XI |
Mathematics |
Mathematics |
9 |
θ = tan ( 180° – ∠ MPQ) = – tan ∠ MPQ MQ y2− y 1 y 2 y 1 . = − = − = x −x MP x1−x 2 2 1 Consequently, we see that in both the cases the slope m of ... |
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| Class XI |
Mathematics |
Mathematics |
9 |
to each other. 1 2 Hence, two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other... |
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| Class XI |
Mathematics |
Mathematics |
9 |
the lines L an1 L (Fig29.6). Then θ = α 2 α an1 α , α 1 902. Therefore tan θ = tan (α – α ) = tan α2 −tan α 1 = m 2m 1 (as 1 + m m ≠ 0) 2 1 1 +an α... |
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| Class XI |
Mathematics |
Mathematics |
9 |
tan = 2 or 1= 2 , 4 1 1 1+ 2 1+ 2 1 1 m − 2 m − 2 =1 or = –1. whichgives 1+ 1 m 1+ 1 m 2 2 Reprint 2026-27 158 MATHEMATICS Therefore m =3 or m = − ... |
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| Class XI |
Mathematics |
Mathematics |
9 |
show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle. ° 7. Findtheslopeoftheline,whichmakesanangleof30 with... |
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| Class XI |
Mathematics |
Mathematics |
9 |
– b [Fig 9.8(b)]. Reprint 2026-27 160 MATHEMATICS Fig 9.8 Example 4 Find the equations of the lines parallel to axes and passing through (– 2, 3). ... |
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