Number Series

Series 1 + 3 + 5 + ... is sum of first n odd numbers. Formula for sum of first n odd numbers = n^2. Setting n^2 = 123454321, we get n = sqrt(123454321) = 11111 (since 11111^2 = 123454321). Therefore, 11111 terms are needed.

The pattern divides by decreasing decimal numbers: 51975 ÷ 5.5 = 9450, 9450 ÷ 4.5 = 2100, 2100 ÷ 3.5 = 600, 600 ÷ 2.5 = 240, 240 ÷ 1.5 = 160. Following this pattern, 160 ÷ 0.5 = 320. Each step reduces the divisor by 1.

The pattern multiplies by 5 and alternates adding/subtracting 4: 6 × 5 - 4 = 26, 26 × 5 + 4 = 134, 134 × 5 - 4 = 666, 666 × 5 + 4 = 3334, 3334 × 5 - 4 = 16666. Continuing: 16666 × 5 + 4 = 83334. This exact value is not listed in options A-D.

Find the pattern in the number series 465, 633, 775, 897, 993, 1065, 1113. Calculate differences: 168, 142, 122, 96, 72, 48. Then second differences: -26, -20, -26, -24, -24. The pattern breaks at 775. The series should follow: starting from 465, add decreasing differences that follow a pattern. Checking: if 775 were correct, the pattern would be inconsistent. The correct number should be approximately 773 or another value that maintains the pattern.

Analyze the pattern in 12, 12, 30, 120, 654, 4620. Check ratios: 12/12=1, 30/12=2.5, 120/30=4, 654/120=5.45, 4620/654≈7.06. The multipliers should follow a pattern. If the pattern were ×1, ×2.5, ×4, ×5.5, ×7, then: 12×1=12, 12×2.5=30, 30×4=120, 120×5.5=660 (not 654), 660×7=4620. So 654 is the wrong number; it should be 660.

The pattern in the series is: each term = (previous term)/2 - 1. Starting from 4866: 4866/2 - 1 = 2433 (but 2432 is given, which is 1 less). For 2432: 2432/2 - 1 = 1215 (but 1218 is given, which is 3 more). From 1218 onwards, the pattern fits perfectly: 1218/2 - 1 = 609 (close to 610), 610/2 - 1 = 304 (close to 306), 306/2 - 1 = 152 (close to 154), 154/2 - 1 = 76 (close to 78). The number 2432 breaks the pattern as it creates an irregularity that propagates through the series.

The pattern follows: multiply by 1 and add 1, multiply by 2 and add 2, multiply by 3 and add 3, etc. 77×1+1=78, 78×2+3=159, 159×3-5=472 (here we should add 4: 159×3+4=481, but 472 is given). The correct term after 159 should be 481 (159×3+4), not 472. Continuing: 472×4-1=1887 (should be 472×4+5=1893), 1889×5+1=9446, 9446×6-1=56675 (should be 9446×6+6=56682). The number 159 disrupts the pattern.

Series 1: 1130-15=1115, 1115+20=1135, 1135-25=1110, 1110+30=1140, so J=1140-35=1105. Series 2: 1=1^1, 2=2^1, 9=3^2, 64=4^3, 625=5^4, so N=6^5=7776. Series 3: differences are 5, 10, 17, 26 (second differences 5, 7, 9), so next difference=37 and L=60+37=97. Then (J+N)-L=(1105+7776)-97=8784.

Find differences: 277-263=14, 290-277=13, 302-290=12, 313-302=11. The pattern is decreasing by 1: +14, +13, +12, +11. Next term = 313 + 10 = 323. Option A is correct. Options B, C, D don't follow this pattern.

The pattern adds 4.4 to each term consistently. After 15.2, the next term should be 15.2+4.4=19.6, not 18.6. Hence 18.6 is the wrong number that breaks the pattern.

The differences should be squares of consecutive odd numbers starting from 5²: 25, 81, 121, 169, 225, 289. The first difference is 35 instead of 25, making 5541 the wrong number (it should be 5531 to maintain the pattern).

श्रृंखला में प्रत्येक पद = पिछला पद × (बढ़ता गुणक) - (घटती संख्या)। 27×2-4=50, 50×3-(?), 50×3-58=92 ≠ 192। सही संबंध: 27×1+23=50, 50×2+92=192, 192×6-12=1140, 1140×8-16=9104, 9104×10-20=91020। पैटर्न: गुणा (1,2,6,8,10) और जोड़ (23,92,-12,-16,-20)। 91020 विकल्प B में है।

श्रृंखला का अंतर बढ़ रहा है: 7-4=3, 13-7=6, 23-13=10, 38-23=15, 59-38=21। अंतर में वृद्धि: 3→6 (+3), 6→10 (+4), 10→15 (+5), 15→21 (+6)। अगला अंतर 21+7=28 होगा। अगला पद = 59+28=87। यह द्वितीय अंतर का समान्य श्रृंखला है जहां प्रत्येक चरण में 1 जुड़ता है। विकल्प C सही है।

The series follows the pattern: ×1 + 5, ×2 + 10, ×3 + 15, ×4 + 20. Applying this: 6 × 1 + 5 = 11, 11 × 2 + 10 = 32, 32 × 3 + 15 = 111, 111 × 4 + 20 = 464. The next term is 464 × 5 + 25 = 2320 + 25 = 2345. The multiplier increases by 1 each step, and the addend increases by 5.

The series is 8, 15, 28, 49, 80, ?. Analyzing the differences: 15-8=7, 28-15=13, 49-28=21, 80-49=31. These differences themselves have a pattern: 13-7=6, 21-13=8, 31-21=10. The second differences increase by 2. Following this pattern, the next second difference is 12, so the next first difference is 31+12=43. Therefore, the next term is 80+43=123.