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A sequence is an arrangement of numbers or objects in a particular format or set of rules. Series is the sum of all the terms of a sequence. Sequences are of two types-\u00a0<\/span><\/b><\/p>\n (1) Infinite terms sequence<\/strong> – When the number of terms is not known or infinite.<\/span><\/b><\/p>\n (2) Finite terms sequence<\/strong> – When the number of terms is known or finite.<\/span><\/b><\/p>\n We can understand this with an example. 2, 4, 6, 8, 10, 12, … is a sequence.<\/span><\/p>\n If we add the numbers in the sequence like 2 + 4 + 6 + 8 + 10 +… this will make a series of the above sequence.<\/span><\/b><\/p>\n Please note that the series can also be finite or infinite depending upon the type of sequence.<\/em><\/strong><\/p>\n <\/span><\/p>\n Some differences between sequence and series are explained below:<\/span><\/p>\n <\/span><\/p>\n There are many types of sequences and series. Some of them are:<\/span><\/b><\/p>\n Let’s discuss these in detail.<\/p>\n In this sequence, every term is obtained by adding or subtracting a particular number from the preceding (previous) number.<\/p>\n For example, 1, 5, 9, 13, … is an arithmetic sequence since every term is obtained by adding \u201c4\u201d to the preceding number. Therefore, 4 is a common difference.<\/p>\n A series obtained by the sum of the elements of an arithmetic sequence is known as the arithmetic series. For example 1 + 5 + 9 + 13… is an arithmetic series.<\/p>\n <\/p>\n In this sequence, every term is obtained by multiplying or dividing a particular number from the preceding number.\u00a0<\/span><\/b><\/span><\/p>\n For example, 1, 3, 9, 27, … is an arithmetic sequence since every term is obtained by multiplying 3 to the preceding number. Therefore, 3 is the common ratio.<\/span><\/b><\/p>\n Similarly as arithmetic series, 1 + 3 + 9 + 27… is a geometric series.<\/span><\/p>\n <\/span><\/p>\n If the reciprocals of all the numbers of the sequence form an arithmetic sequence, then such a sequence is called a harmonic sequence.\u00a0<\/span><\/b><\/p>\n For example, 1,\\frac{1}{3},\\frac{1}{6}, \\frac{1}{9}<\/span> ,… is a harmonic sequence, and 1+\\frac{1}{3} + \\frac{1}{6} + \\frac{1}{9}<\/span>…. is a harmonic series.<\/span><\/p>\n <\/span><\/p>\n There are many formulas for different sequences and series. Using these formulas, we can find unknown values like the first term, n<\/span>th<\/span> term, common parameters, etc.\u00a0<\/span><\/p>\n <\/span><\/p>\n Arithmetic Sequence and Series Formula<\/b><\/b><\/p>\n We use various formulas in arithmetic sequence as given below:<\/span><\/b><\/p>\n The arithmetic sequence is represented by a, a + d, a + 2d, a + 3d, \u2026<\/span><\/p>\n So, the arithmetic series will be a + (a + d) + (a + 2d) + (a + 3d) + \u2026<\/span><\/b><\/p>\n Here, a = first term and <\/span>d = common difference (successive term – preceding term).<\/span><\/p>\n The formula for \\mathrm{n}^{\\text {th }}<\/span><\/span>term of the sequence = a + (n – 1)d<\/span><\/p>\n Sum of arithmetic series \\left(S_{n}\\right)=\\frac{n}{ 2}[(2 a+(n-1) d)]<\/span><\/span><\/p>\n <\/p>\n Geometric sequence is given by a, a r, a r^{2}, \\ldots, a r^{(n-1)}, \\ldots<\/span><\/span><\/b><\/p>\n So, geometric series will be a+a r+a r^{2}+\\ldots+a r^{(n-1)}+\\ldots<\/span><\/span><\/p>\n Here, a = first term and <\/span>r = common ration (successive term\/preceding term).<\/span><\/p>\n The formula for\\mathrm{n}^{\\text {th }}<\/span> term a= r^{(n-1)}<\/span><\/span><\/p>\n Sum of geometric series \\left(\\mathrm{S}_{n}\\right):<\/span><\/span><\/p>\n If r<1, S_{n}=a\\left(1-r^{n}\\right) \/(1-r) <\/span><\/span><\/p>\n and when r>1, S_{n}=a\\left(r^{n}-1\\right) \/(r-1)<\/span>. <\/span><\/p>\n Solution:<\/strong><\/p>\n Given a = -3,\u00a0<\/span><\/p>\n \u00a0d =\\frac {-5}{2}-{(-3)} = \\frac{-5}{2}+3 = \\frac{1}{2}, <\/span><\/span><\/span><\/p>\n \u00a0n = 12 Formula for the<\/span> \\mathrm{n}^{\\text {th }}<\/span> <\/span><\/span>term of an arithmetic sequence:<\/span><\/p>\n \\therefore \\mathrm{12}^{\\text {th }}<\/span>term <\/span>= -3 + [(12-1) \\times \\frac{1}{2})]<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= -3 + (11 \\times \\frac{1}{2})<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= -3 +\\frac{11}{2}<\/span><\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=\\frac{5}{2}<\/span><\/span><\/p>\n Hence, the \\mathrm{12}^{\\text {th }}<\/span>term of the given sequence is \\frac{5}{2}<\/span>.<\/span><\/p>\n <\/span><\/p>\n Given: a = 1, r =\\frac{1}{4}\/1 = \\frac{1}{4}<\/span>.<\/span><\/p>\n Formula for the \\mathrm{n}^{\\text {th }}<\/span>term of a geometric sequence:<\/span><\/p>\n \\mathrm{4}^{\\text {th }}<\/span>term =a r^{(n-1)}<\/span><\/span><\/p>\n \\mathrm{5}^{\\text {th }}<\/span>term =1 \\times \\left(\\frac{1} { 4}\\right)^{(5-1)} <\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\left(\\frac{1}{ 4}\\right)^{4} <\/span> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\frac{1}{256}<\/span><\/span><\/p>\n Therefore, the 5th term of the given sequence is \\frac {1}{256}<\/span>. <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Row for space” _builder_version=”4.10.6″ _module_preset=”default” locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.11″ _module_preset=”default” global_colors_info=”{}”][et_pb_divider show_divider=”off” _builder_version=”4.10.4″ _module_preset=”default” global_colors_info=”{}”][\/et_pb_divider][\/et_pb_column][\/et_pb_row][\/et_pb_section][et_pb_section fb_built=”1″ admin_label=”banner and faq Section” module_class=”mainsec2″ _builder_version=”4.10.4″ _module_preset=”default” custom_padding=”40px||0px||false|false” global_colors_info=”{}”][et_pb_row use_custom_gutter=”on” gutter_width=”1″ make_equal=”on” disabled_on=”on|on|off” admin_label=”banner Row” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fff7d6″ width=”100%” max_width=”1310px” height=”134px” custom_margin=”||50px||false|false” custom_padding=”12px||12px||true|false” border_radii=”on|11px|11px|11px|11px” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_image src=”https:\/\/stgwebsite.mindspark.in\/wordpress\/wp-content\/uploads\/2021\/08\/calloutImage.png” title_text=”calloutImage” show_bottom_space=”off” admin_label=”Image” module_class=”img1″ _builder_version=”4.10.8″ _module_preset=”default” width=”25px” height=”60px” custom_padding=”2px||2px||true|false” global_colors_info=”{}”][\/et_pb_image][et_pb_text module_class=”ftstyle” _builder_version=”4.9.10″ _module_preset=”default” text_orientation=”center” global_colors_info=”{}”]<\/p>\nDifference Between Sequence and Series<\/b><\/h2>\n
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<\/span><\/li>\n
<\/span><\/li>\nTypes of Sequence and Series<\/b><\/h2>\n
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Arithmetic Sequence and Series<\/b><\/h3>\n
Geometric Sequence and Series<\/b><\/h3>\n
Harmonic Sequence and Series<\/b><\/h3>\n
Sequence and Series Formulas<\/b><\/h2>\n
Geometric Sequence and Series Formulas<\/b><\/h3>\n
<\/span><\/span><\/p>\nExamples<\/b><\/h2>\n
1. What will be the \\mathrm{12}^{\\text {th }}<\/span><\/span>term of the arithmetic sequence <\/span><\/h3>\n
-3, \\frac{-5}{2}<\/span>, -2\u2026.?<\/span><\/span><\/h3>\n
<\/span><\/p>\n \\mathrm{n}^{\\text {th }}<\/span> term = a + (n-1)d<\/h3>\n
2. Using sequence formula, find the next term of the given geometric sequence: 1, \\frac{1}{4}, \\frac{1}{16}, \\frac{1}{64}<\/span> \u2026<\/span><\/h3>\n
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<\/span><\/p>\n
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\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n